The On-Line Encyclopedia of Integer Sequences (OEIS)

Revisions by Peter Luschny

(See also Peter Luschny's wiki page
and changes approved by Peter Luschny)

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A034869

Right half of Pascal's triangle.
(history; published version)

#44 by Peter Luschny at Tue Sep 24 15:54:30 EDT 2024

STATUS

reviewed

approved

A376072

allocated for Mingjian Ding
(history; published version)

#31 by Peter Luschny at Tue Sep 24 03:00:39 EDT 2024

STATUS

editing

proposed

A375853

Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.
(history; published version)

#37 by Peter Luschny at Tue Sep 24 03:00:33 EDT 2024

STATUS

editing

proposed

A376072

allocated for Mingjian Ding
(history; published version)

#30 by Peter Luschny at Tue Sep 24 03:00:16 EDT 2024

MAPLE

# Assuming offset 2:

A375853

Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.
(history; published version)

#36 by Peter Luschny at Tue Sep 24 02:25:13 EDT 2024

MAPLE

# Alternatively, as coefficients using the generating function of the row polynomials:

mpa rgf := (n, x) -> ((sqrt(x) - 1)^(2*n)*(2*n*sqrt(x) + x + 1) - (sqrt(x) + 1)^(2*n)*(-2*n*sqrt(x) + x + 1))/(16*sqrt(x)):

T := (n, k) -> coeff(expand(mpargf(n, x)), x, k):

STATUS

proposed

editing

A055584

Triangle of partial row sums (prs) of triangle A055252.
(history; published version)

#8 by Peter Luschny at Mon Sep 23 16:31:58 EDT 2024

STATUS

editing

proposed

A055252

Triangle of partial row sums (prs) of triangle A055249.
(history; published version)

#8 by Peter Luschny at Mon Sep 23 16:31:48 EDT 2024

STATUS

editing

proposed

A034869

Right half of Pascal's triangle.
(history; published version)

#43 by Peter Luschny at Mon Sep 23 16:31:14 EDT 2024

STATUS

proposed

reviewed

A244823

Decimal expansion of c/P in SI units (meters), where c is the speed of light in vacuum and P is the number of periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom in one second.
(history; published version)

#26 by Peter Luschny at Mon Sep 23 16:25:16 EDT 2024

STATUS

proposed

reviewed

A055252

Triangle of partial row sums (prs) of triangle A055249.
(history; published version)

#7 by Peter Luschny at Mon Sep 23 14:38:15 EDT 2024

FORMULA

T(n, k) = binomial(n, k)*hypergeom([3, k - n], [k + 1], -1). - Peter Luschny, Sep 23 2024

EXAMPLE

[0] 1

[1] 4, 1

[2] 13, 5, 1

{1}; {4,1}; {13,5,1}; {[3] 38, 18, 6, 1};...

[4] 104, 56, 24, 7, 1

[5] 272, 160, 80, 31, 8, 1

[6] 688, 432, 240, 111, 39, 9, 1

[7] 1696, 1120, 672, 351, 150, 48, 10, 1

Fourth row polynomial (n = 3): p(3, x) = 38 + 18*x + 6*x^2 + x^3.

MAPLE

T := (n, k) -> binomial(n, k)*hypergeom([3, k - n], [k + 1], -1):

for n from 0 to 7 do seq(simplify(T(n, k)), k = 0..n) od; # Peter Luschny, Sep 23 2024

STATUS

approved

editing