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A143382 -id:A143382 - OEIS
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A143383
Denominator of Sum_{k=0..n} 1/k!!.
+10
9
1, 1, 2, 6, 24, 40, 240, 560, 13440, 120960, 241920, 887040, 394240, 138378240, 276756480, 593049600, 66421555200, 4136140800, 173717913600, 14302774886400, 171633298636800, 144171970854912, 7208598542745600, 283414985441280
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Numerators are A143382. A143382(n)/A143383(n) is to A007676(n)/A007676(n) as double factorials are to factorials. A143382/A143383 fractions begin:
n numerator/denominator
0 1/0!! = 1/1
1 1/0!! + 1/1!! = 2/1
2 1/0!! + 1/1!! + 1/2!! = 5/2
3 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6
4 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! = 71/24
5 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! = 121/40
6 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! = 731/240
The series converges to sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) = 3.0594074053425761445... whose decimal expansion is given by A143280. The analogs of A094007 and A094008 are determined by 2 being the only prime denominator in the convergents to the sum of reciprocals of double factorials and prime numerators beginning: a(1) = 2, a(2) = 5, a(3) = 17, a(4) = 71, a(15) = 1814380259, a(19) = 43758015399281, a(21) = 441080795274037, a(23) = 867081905243923.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
Eric W. Weisstein, Double Factorial. Gives formula for limit of series, which was independently derived by Max Alekseyev.
FORMULA
Denominators of Sum_{k=0..n} 1/k!! = Sum_{k=0..n} 1/A006882(k).
EXAMPLE
a(3) = 6 because 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6.
a(15) = 593049600 because 1814380259/593049600 = 1/1 + 1/1 + 1/2 + 1/3 + 1/8 + 1/15 + 1/48 + 1/105 + 1/384 + 1/945 + 1/3840 + 1/10395 + 1/46080 + 1/135135 + 1/645120 + 1/2027025.
MATHEMATICA
Table[Denominator[Sum[1/k!!, {k, 0, n}]], {n, 0, 25}] (* G. C. Greubel, Mar 28 2019 *)
PROG
(PARI) vector(25, n, n--; denominator(sum(k=0, n, 1/prod(j=0, floor((k-1)/2), (k - 2*j)) ))) \\ G. C. Greubel, Mar 28 2019
(Magma) [n le 0 select 1 else Denominator( 1 + (&+[ 1/(0 + (&*[k-2*j: j in [0..Floor((k-1)/2)]])) : k in [1..n]]) ): n in [0..25]]; // G. C. Greubel, Mar 28 2019
(Sage) [denominator(sum(1/product((k-2*j) for j in (0..floor((k-1)/2))) for k in (0..n))) for n in (0..25)] # G. C. Greubel, Mar 28 2019
CROSSREFS
Cf. A006882 (n!!), A094007, A143280 (m(2)), A143382 (numerator).
KEYWORD
easy,frac,nonn
AUTHOR
Jonathan Vos Post, Aug 11 2008
STATUS
approved
A354211
a(n) is the numerator of Sum_{k=0..n} 1 / (2*k+1)!.
+10
5
1, 7, 47, 5923, 426457, 15636757, 7318002277, 1536780478171, 603180793741, 142957467201379447, 60042136224579367741, 10127106976545720025649, 18228792557782296046168201, 12796612375563171824410077103, 3463616416319098507140327535879, 1380498543075754976417359117871773
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
FORMULA
Numerators of coefficients in expansion of sinh(sqrt(x)) / (sqrt(x) * (1 - x)).
EXAMPLE
1, 7/6, 47/40, 5923/5040, 426457/362880, 15636757/13305600, 7318002277/6227020800, ...
MATHEMATICA
Table[Sum[1/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Sinh[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Numerator
PROG
(PARI) a(n) = numerator(sum(k=0, n, 1/(2*k+1)!)); \\ Michel Marcus, May 24 2022
(Python)
from fractions import Fraction
from math import factorial
def A354211(n): return sum(Fraction(1, factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022
CROSSREFS
Cf. A009445, A053557, A061354, A073742, A103816, A120265, A143382, A289381, A354331 (denominators), A354332, A354334.
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, May 24 2022
STATUS
approved
A354332
a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k+1)!.
+10
5
1, 5, 101, 4241, 305353, 33588829, 209594293, 1100370038249, 23023126954133, 102360822438075317, 42991545423991633141, 4350744396907953273869, 13052233190723859821607001, 9162667699888149594768114701, 7440086172309177470951709137213, 364172638960396581472899447242531
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
FORMULA
Numerators of coefficients in expansion of sin(sqrt(x)) / (sqrt(x) * (1 - x)).
EXAMPLE
1, 5/6, 101/120, 4241/5040, 305353/362880, 33588829/39916800, 209594293/249080832, ...
MATHEMATICA
Table[Sum[(-1)^k/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Sin[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Numerator
PROG
(PARI) a(n) = numerator(sum(k=0, n, (-1)^k/(2*k+1)!)); \\ Michel Marcus, May 24 2022
(Python)
from fractions import Fraction
from math import factorial
def A354332(n): return sum(Fraction(-1 if k % 2 else 1, factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022
CROSSREFS
Cf. A009445, A049469, A053557, A061354, A103816, A120265, A143382, A354211, A354298, A354333 (denominators), A354334.
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, May 24 2022
STATUS
approved
A354334
a(n) is the numerator of Sum_{k=0..n} 1 / (2*k)!.
+10
5
1, 3, 37, 1111, 6913, 799933, 739138093, 44841044309, 32285551902481, 9879378882159187, 1251387991740163687, 1734423756551866870183, 136771701945232930334431, 23048564587067030852654113, 42769754577382930342215977687, 409306551305554643375006906464591
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
FORMULA
Numerators of coefficients in expansion of cosh(sqrt(x)) / (1 - x).
EXAMPLE
1, 3/2, 37/24, 1111/720, 6913/4480, 799933/518400, 739138093/479001600, ...
MATHEMATICA
Table[Sum[1/(2 k)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Cosh[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator
Accumulate[1/(2*Range[0, 20])!]//Numerator (* Harvey P. Dale, Sep 05 2024 *)
PROG
(PARI) a(n) = numerator(sum(k=0, n, 1/(2*k)!)); \\ Michel Marcus, May 24 2022
(Python)
from fractions import Fraction
from math import factorial
def A354334(n): return sum(Fraction(1, factorial(2*k)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022
CROSSREFS
Cf. A010050, A053557, A061354, A073743, A103816, A120265, A143382, A354211, A354332, A354335 (denominators).
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, May 24 2022
STATUS
approved
A354298
a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.
+10
4
1, 2, 11, 76, 137, 7534, 97943, 1469144, 24975449, 94906706, 9965204131, 229199695012, 5729992375301, 9100576125478, 897316805972131, 563093542209232, 4589775462547450033, 5539384178936577626, 5943759223998947792699, 46361321947191792783052, 9504070999174317520525661
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Robert Israel, Table of n, a(n) for n = 1..403
FORMULA
Numerators of coefficients in expansion of sqrt(Pi*x*exp(-x)/2) * erfi(sqrt(x/2)) / (1 - x).
EXAMPLE
1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...
MAPLE
S:= 0: R:= NULL:
for n from 1 to 100 do
S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
R:= R, numer(S);
od:
R; # Robert Israel, Jan 10 2024
MATHEMATICA
Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Numerator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Numerator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Numerator
CROSSREFS
Cf. A001147, A053557, A061354, A064646, A103816, A113012, A120265, A143382, A289381, A306858, A354299 (denominators).
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, May 23 2022
STATUS
approved
A354302
a(n) is the numerator of Sum_{k=0..n} 1 / (k!)^2.
+10
2
1, 2, 9, 41, 1313, 5471, 1181737, 28952557, 1235309099, 150090055529, 30018011105801, 201787741322329, 523033825507476769, 44196358255381786981, 5774990812036553498851, 1949059399062336805862213, 997918412319916444601453057, 3697415655903280160125896583
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..17.
FORMULA
Numerators of coefficients in expansion of BesselI(0,2*sqrt(x)) / (1 - x).
EXAMPLE
1, 2, 9/4, 41/18, 1313/576, 5471/2400, 1181737/518400, 28952557/12700800, 1235309099/541900800, ...
MATHEMATICA
Table[Sum[1/(k!)^2, {k, 0, n}], {n, 0, 17}] // Numerator
nmax = 17; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator
CROSSREFS
Cf. A001044, A006040, A053557, A061354, A070910, A103816, A120265, A143382, A354303 (denominators), A354304.
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, May 23 2022
STATUS
approved
A354304
a(n) is the numerator of Sum_{k=0..n} (-1)^k / (k!)^2.
+10
2
1, 0, 1, 2, 43, 403, 23213, 118483, 51997111, 1842647621, 327581799289, 8918414485643, 4670006130663971, 361730891537680087, 130890931830249779173, 427294615628884602769, 6534075316966068976316143, 885163015595247156635327497, 41526561745210509140249210357
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..18.
FORMULA
Numerators of coefficients in expansion of BesselJ(0,2*sqrt(x)) / (1 - x).
EXAMPLE
1, 0, 1/4, 2/9, 43/192, 403/1800, 23213/103680, 118483/529200, 51997111/232243200, 1842647621/8230118400, ...
MATHEMATICA
Table[Sum[(-1)^k/(k!)^2, {k, 0, n}], {n, 0, 18}] // Numerator
nmax = 18; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator
CROSSREFS
Cf. A001044, A053557, A061354, A073701, A091681, A103816, A120265, A143382, A354302, A354305 (denominators).
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, May 23 2022
STATUS
approved
A354138
a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k)!.
+10
1
1, 1, 13, 389, 4357, 1960649, 258805669, 47102631757, 11304631621681, 691843455246877, 1314502564969066301, 607300185015708631061, 335229702128671164345673, 217899306383636256824687449, 32946375125205802031892742289, 848027998784883070051677094421
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..15.
FORMULA
Numerators of coefficients in expansion of cos(sqrt(x)) / (1 - x).
EXAMPLE
1, 1/2, 13/24, 389/720, 4357/8064, 1960649/3628800, 258805669/479001600, ...
MATHEMATICA
Table[Sum[(-1)^k/(2 k)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Cos[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator
PROG
(PARI) a(n) = numerator(sum(k=0, n, (-1)^k/(2*k)!)); \\ Michel Marcus, May 24 2022
CROSSREFS
Cf. A010050, A049470, A053557, A061354, A103816, A120265, A143382, A354211, A354332, A354334, A354378 (denominators).
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, May 24 2022
STATUS
approved
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