OFFSET
0,2
COMMENTS
Coefficients in a simultaneous approximation to zeta(2) and zeta(3).
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, Sep 03 2004
STATUS
approved