A104684 -id:A104684 - cp's OEIS frontend

A171485 Beukers integral Integral_{y = 0..1} Integral_{x = 0..1} -log(xy) / (1-xy) * P_n(2x-1) P_n(2y-1) dx dy = (A(n) + B(n)zeta(3)) / A003418(n)^3. This sequence gives the values of B(n).

2, 10, 1168, 624240, 114051456, 353810160000, 9271076400000, 86580328116240000, 19402654331894400000, 15000926812307614080000, 437120128035736887168000, 17196604114594832318160000000, 514325437537328572480262784000, 34134351456507030556755674947200000

Offset: 0

Max Alekseyev, Dec 09 2009

Values of A(n) are given in A171484. P_n(x) are the Legendre Polynomials defined by n!*P_n(x) = (d/dx)^n (x^n*(1-x)^n), see A008316.

F. Beukers, A note on the irrationality of zeta(2) and zeta(3), Bull. London Math. Soc., Vol. 11, No. 3 (1979), 268-272.
Wikipedia, Apéry's theorem

Cf. A002117, A003418, A005259, A104684.

Cf. A008316, A171484.

seq( 2 * lcm(seq(i, i = 1..n))^3 * add(binomial(n,k)^2*binomial(n+k,k)^2, k = 0..n), n = 0..20); # Peter Bala, Aug 01 2025

Join[{2}, Table[2*(LCM @@ Range[n])^3 * HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1, 1, 1}, 1], {n, 1, 20}]] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = 2 * A003418(n)^3 * A005259(n). - Peter Bala, Aug 01 2025

A092371 Triangle read by rows: T(n, k) = binomial(n, k) * binomial(n+k, n-k).

Original entry on oeis.org

1, 6, 1, 18, 15, 1, 40, 90, 28, 1, 75, 350, 280, 45, 1, 126, 1050, 1680, 675, 66, 1, 196, 2646, 7350, 5775, 1386, 91, 1, 288, 5880, 25872, 34650, 16016, 2548, 120, 1, 405, 11880, 77616, 162162, 126126, 38220, 4320, 153, 1, 550, 22275, 205920, 630630

Offset: 1

Benoit Cloitre, Mar 20 2004

Related to the coefficients of x^k y^k in the n-th power of x^2 + x*y + 2*x + y + 1. - F. Chapoton, Jan 04 2025

			Triangle starts:
  [1]   1;
  [2]   6,    1;
  [3]  18,   15,     1;
  [4]  40,   90,    28,     1;
  [5]  75,  350,   280,    45,     1;
  [6] 126, 1050,  1680,   675,    66,    1;
  [7] 196, 2646,  7350,  5775,  1386,   91,   1;
  [8] 288, 5880, 25872, 34650, 16016, 2548, 120, 1;

First column = A002411, second column = A001297, third column = A107418, fourth column = A105251, fifth column = A104673.
Main diagonal = 1, second diagonal = A000384.
Cf. A063007, A006480 (central terms), A082759 (row sums + 1).
Cf. A104684.

T := (n, k) -> binomial(n, k) * binomial(n+k, n-k):  # Peter Luschny, Jan 04 2025

T(n,k) = binomial(n,k)*binomial(n+k,n-k)

T(n, k) = [x^(n-k)] F(-n, -n-k; 1; x). - Paul Barry, Sep 04 2008

cp's OEIS Frontend

A171485 Beukers integral Integral_{y = 0..1} Integral_{x = 0..1} -log(xy) / (1-xy) * P_n(2x-1) P_n(2y-1) dx dy = (A(n) + B(n)zeta(3)) / A003418(n)^3. This sequence gives the values of B(n).

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A092371 Triangle read by rows: T(n, k) = binomial(n, k) * binomial(n+k, n-k).

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cp's OEIS Frontend

A171485 Beukers integral Integral_{y = 0..1} Integral_{x = 0..1} -log(x*y) / (1-x*y) * P_n(2*x-1) * P_n(2*y-1) dx dy = (A(n) + B(n)*zeta(3)) / A003418(n)^3. This sequence gives the values of B(n).

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A092371 Triangle read by rows: T(n, k) = binomial(n, k) * binomial(n+k, n-k).

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Maple

PARI

Formula

A171485 Beukers integral Integral_{y = 0..1} Integral_{x = 0..1} -log(xy) / (1-xy) * P_n(2x-1) P_n(2y-1) dx dy = (A(n) + B(n)zeta(3)) / A003418(n)^3. This sequence gives the values of B(n).