OFFSET
0,2
COMMENTS
The Euler-Seidel matrix for the sequence {k!} is array A076571 read as a square, whose k-th column entries have a common factor of k!. Removing these common factors gives the current table.
This table is closely connected to the constant 1/e. The row, column and diagonal entries of this table occur in series acceleration formulas for 1/e.
LINKS
Robert Israel, Table of n, a(n) for n = 0..10010 (antidiagonals 0 to 140, flattened)
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Eric Weisstein's World of Mathematics Poisson-Charlier polynomial
FORMULA
T(n,k) = (1/k!)*Sum_{j = 0..n} binomial(n,j)*(k+j)!.
T(n,k) = ((n+k)!/k!)*Num_Pade(n,k), where Num_Pade(n,k) denotes the numerator of the Padé approximation for the function exp(x) of degree (n,k) evaluated at x = 1.
Recurrence relations:
T(n,k) = T(n-1,k) + (k+1)*T(n-1,k+1);
T(n,k) = (n+k)*T(n-1,k) + T(n-1,k-1).
E.g.f. for column k: exp(y)/(1-y)^(k+1).
E.g.f. for array: exp(y)/(1-x-y) = (1 + x + x^2 + ...) + (2 + 3*x + 4*x^2 + ...)*y + (5 + 11*x + 19*x^2 + ...)*y^2/2! + ... .
Row n lists the values of the Poisson-Charlier polynomial x^(n) + C(n,1)*x^(n-1) + C(n,2)*x^(n-2) + ... + C(n,n) for x = 1,2,3,..., where x^(m) denotes the rising factorial x*(x+1)*...*(x+m-1).
Main diagonal is A001517.
Series formulas for 1/e:
Row n: 1/e = n!*[1/T(n,0) - 1/(1!*T(n,0)*T(n,1)) + 1/(2!*T(n,1)*T(n,2)) - 1/(3!*T(n,2)*T(n,3)) + ...].
Column k: k!/e = A000166(k) + (-1)^(k+1)*[0!/(T(0,k)*T(1,k)) + 1!/(T(1,k)*T(2,k)) + 2!/(T(2,k)*T(3,k)) + ...].
Main diagonal: 1/e = 1 - 2*Sum_{n>=0} (-1)^n/(T(n,n)*T(n+1,n+1)) = 1 - 2*[1/(1*3) - 1/(3*19) + 1/(19*193) - ...].
Second subdiagonal: 1/e = 2*(1^2/(1*5) - 2^2/(5*49) + 3^2/(49*685) - ...).
Compare with A143413.
From Peter Luschny, Oct 05 2017: (Start)
T(n, k) = hypergeom([k+1, k-n], [], -1).
EXAMPLE
The Euler-Seidel matrix for the sequence {k!} begins
==============================================
n\k|.....0.....1.....2.....3.....4.....5.....6
==============================================
0..|.....1.....1.....2.....6....24...120...720
1..|.....2.....3.....8....30...144...840
2..|.....5....11....38...174...984
3..|....16....49...212..1158
4..|....65...261..1370
5..|...326..1631
6..|..1957
...
Dividing the k-th column by k! gives
==============================================
n\k|.....0.....1.....2.....3.....4.....5.....6
==============================================
0..|.....1.....1.....1.....1.....1.....1.....1
1..|.....2.....3.....4.....5.....6.....7
2..|.....5....11....19....29....41
3..|....16....49...106...193
4..|....65...261...685
5..|...326..1631
6..|..1957
...
Examples of series formula for 1/e:
Row 2: 1/e = 2*(1/5 - 1/(1!*5*11) + 1/(2!*11*19) - 1/(3!*19*29) + ...).
Column 4: 24/e = 9 - (0!/(1*6) + 1!/(6*41) + 2!/(41*316) + ...).
...
Displayed as a triangle:
0 | 1
1 | 2, 1
2 | 5, 3, 1
3 | 16, 11, 4, 1
4 | 65, 49, 19, 5, 1
5 | 326, 261, 106, 29, 6, 1
6 | 1957, 1631, 685, 193, 41, 7, 1
7 | 13700, 11743, 5056, 1457, 316, 55, 8, 1
MAPLE
T := (n, k) -> 1/k!*add(binomial(n, j)*(k+j)!, j = 0..n):
for n from 0 to 9 do seq(T(n, k), k = 0..9) end do;
# Alternate:
T:= proc(n, k) option remember;
if n = 0 then return 1 fi;
(n+k)*procname(n-1, k) + procname(n-1, k-1);
end proc:
seq(seq(T(s-n, n), n=0..s), s=0..10); # Robert Israel, Jul 07 2017
# Or:
A143409 := (n, k) -> hypergeom([k+1, k-n], [], -1):
seq(seq(simplify(A143409(n, k)), k=0..n), n=0..9); # Peter Luschny, Oct 05 2017
MATHEMATICA
T[n_, k_] := HypergeometricPFQ[{k+1, k-n}, {}, -1];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Peter Luschny, Oct 05 2017 *)
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Aug 14 2008
STATUS
approved