A098274 Sum_{k = 0..n} C(n, k)^2*C(n+k, n)*C(n+2*k, n).

1, 7, 163, 5191, 191251, 7665757, 324610399, 14287393351, 647133545107, 29966682138757, 1412267381766913, 67516263948005341, 3266295503353540399, 159606073670867165713, 7866072852462175900663

Offset

0,2

Comments

Coefficients in a simultaneous approximation to zeta(2) and zeta(3).

Links

Table of n, a(n) for n=0..14.

W. Zudilin, Approximations to -, di- and tri-logarithms, arXiv:math/0409023 [math.CA], 2004-2005.

Formula

a(n) = (-1)^n * Sum_{k = 0..n} (-1)^k*C(n, k)*C(n+k, k)^3.

From Peter Bala, Jan 18 2020: (Start)

a(n) = Sum_{0 <= j, k <= n} (-1)^(j+k)*C(n, k)*C(n, j)*C(n+k, k)*C(n+k+j, k+j)^2.

a(n) = Sum_{0 <= j, k <= n} (-1)^(n+k)*C(n, k)*C(n, j)^2*C(n+k, k)*C(n+k+j, k+j).

(End)

a(n) = hypergeom([1/2+n/2, 1+n/2, -n, -n, 1+n], [1/2, 1, 1, 1], 1). - Peter Luschny, Jan 19 2020

D-finite with recurrence +68*n^3*(2*n-1)*a(n) +2*(-3030*n^4+3575*n^3+1877*n^2-4229*n+1569)*a(n-1) +2*(-38818*n^4+271866*n^3-701843*n^2+794123*n-332895)*a(n-2) +2*(-3877*n^4+51375*n^3-245797*n^2+509352*n-388548)*a(n-3) +(n-3)*(29*n^3-1547*n^2+8736*n-13116)*a(n-4) -48*(n-3)*(n-4)^3*a(n-5)=0. - R. J. Mathar, Jul 27 2022

Maple

a := n -> hypergeom([1/2 + n/2, 1 + n/2, -n, -n, 1 + n], [1/2, 1, 1, 1],  1):
seq(simplify(a(n)), n=0..14); # Peter Luschny, Jan 19 2020

Mathematica

Table[Sum[Binomial[n, k]^2 Binomial[n+k, n]Binomial[n+2k, n], {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Aug 17 2019 *)

Prog

(Magma) [&+[Binomial(n, k)^2 * Binomial(n+k, n) * Binomial(n+2*k, n): k in [0..n]]:n in  [0..15]]; // Marius A. Burtea, Jan 19 2020

Crossrefs

Cf. A005258, A005259, A098275.

Sequence in context: A201179 A218998 A220921 * A364114 A363698 A027549

Adjacent sequences: A098271 A098272 A098273 * A098275 A098276 A098277

Keyword

nonn,easy

Author

Ralf Stephan, Sep 03 2004

Status

approved