A036916 a(n) = Sum_{k=0..n} binomial(2*n-2*k,n-k)^2 * binomial(n,k)^2.

1, 5, 53, 761, 12661, 229705, 4410665, 88127485, 1813270645, 38158684745, 817458330553, 17767242718285, 390819348043369, 8683822363169933, 194618212789162733, 4394243766346694161, 99862206804817230965, 2282427331053360624713

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• Cf. M. Petkovsek et al., A=B, Peters, p. ix.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A036915, A036917.

Cf. A007318.

Row n=4 of A275784.

Programs

Haskell

a036916 n = sum $ map
   (\k -> (a007318 (2*n-2*k) (n-k))^2 * (a007318 n k)^2) [0..n]
-- Reinhard Zumkeller, May 24 2012

Mathematica

Table[Sum[Binomial[2n-2k,n-k]^2 Binomial[n,k]^2,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 31 2013 *)

Formula

(n - 1)*(144*n^3 - 864*n^2 + 1693*n - 1075)*n^3*a(n) - 2*(n - 1)*(2592*n^6 - 19440*n^5 + 56322*n^4 - 80296*n^3 + 60004*n^2 - 23017*n + 3580)*a(n - 1) + (42336*n^7 - 423360*n^6 + 1769838*n^5 - 4006912*n^4 + 5293968*n^3 - 4062414*n^2 + 1661406*n - 274520)*a(n - 2) - 2*(34848*n^5 - 261360*n^4 + 741842*n^3 - 984642*n^2 + 598948*n - 127215)*(n - 2)^2*a(n - 3) + 225*(144*n^3 - 432*n^2 + 397*n - 102)*(n - 2)^2*(n - 3)^2*a(n - 4) = 0 - Vladeta Jovovic, Jul 15 2004

a(n) ~ 5^(2*n+2) / (16 * (Pi*n)^(3/2)). - Vaclav Kotesovec, Mar 02 2014