A092813 - OEIS

%I #41 Feb 20 2025 14:02:40

%S 1,9,433,36729,3824001,450954009,58160561761,7989733343097,

%T 1149808762915201,171540347534028009,26338900959100106433,

%U 4140153621102790276137,663592912043903970182289,108127319237119098011204937,17868369859451104998973346433,2989001418301890511076878884729

%N Schmidt's problem sum for r = 3.

%C 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

%H Vincenzo Librandi, <a href="/A092813/b092813.txt">Table of n, a(n) for n = 0..200</a>

%H S. Hassani, J.-M. Maillard, and N. Zenine, <a href="https://arxiv.org/abs/2502.05543">On the diagonals of rational functions: the minimal number of variables (unabridged version)</a>, arXiv:2502.05543 [math-ph], 2025. See p. 11.

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of generalized Apery sequences with powers of binomial coefficients</a>, Nov 04 2012.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>.

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

%F a(n) ~ (1+sqrt(2))^(3*(2*n+1))/(2^(9/4)*(Pi*n)^(5/2)*sqrt(3)). - _Vaclav Kotesovec_, Nov 04 2012

%t Table[Sum[Binomial[n,k]^3 Binomial[n+k,k]^3,{k,0,n}],{n, 0, 20}] (*_Harvey P. Dale_, Apr 26 2011 *)

%o (PARI) a(n)=sum(k=0,n,binomial(n,k)^3*binomial(n+k,k)^3); \\ _Joerg Arndt_, May 11 2013

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

%K nonn,easy

%O 0,2

%A _Eric W. Weisstein_, Mar 06 2004

%E Prepended missing a(0)=1, _Joerg Arndt_, May 11 2013