A092813 Schmidt's problem sum for r = 3.

1, 9, 433, 36729, 3824001, 450954009, 58160561761, 7989733343097, 1149808762915201, 171540347534028009, 26338900959100106433, 4140153621102790276137, 663592912043903970182289, 108127319237119098011204937, 17868369859451104998973346433, 2989001418301890511076878884729

Offset: 0

Author

Eric W. Weisstein, Mar 06 2004

Keywords

• nonn
• easy

Comments

Apparently, the diagonal of 1/((1 - x - y)*(1 - z - t)*(1 - u - w) - x*y*z*t*u*w). - Peter Bala, Jun 30 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 11.
• Vaclav Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012.
• Eric Weisstein's World of Mathematics, Schmidt's Problem.

Crossrefs

Cf. A001850, A005259, A092814, A092815, A218689.

Programs

Mathematica

Table[Sum[Binomial[n,k]^3 Binomial[n+k,k]^3,{k,0,n}],{n, 0, 20}] (*Harvey P. Dale, Apr 26 2011 *)

PARI

a(n)=sum(k=0,n,binomial(n,k)^3*binomial(n+k,k)^3); \\ Joerg Arndt, May 11 2013

Formula

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

a(n) ~ (1+sqrt(2))^(3*(2*n+1))/(2^(9/4)*(Pi*n)^(5/2)*sqrt(3)). - Vaclav Kotesovec, Nov 04 2012

Extensions

Prepended missing a(0)=1, Joerg Arndt, May 11 2013