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A111968 a(n) = Sum_{k=0..n} C(n,k)^2*C(n+k,k)^3, where C := binomial. 6
1, 9, 325, 17577, 1152501, 84505509, 6664647781, 553268669865, 47710914870133, 4236909872278509, 385139801423145825, 35681384898462925125, 3358273513450241419125, 320308335005997679093125, 30900030366269721747776325, 3010365811746267487293617577 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

FORMULA

a(n) ~ (1+r)^(6*n+7/2)/r^(5*n+9/2)/(4*Pi^2*n^2)*sqrt((1-r)/(5-r)), where r is positive real root of the equation (1-r)^2*(1+r)^3=r^5, r = 0.77591859532439... - Vaclav Kotesovec, Nov 04 2012

Recurrence: (n-1)^2*n^4*(843719*n^6 - 10346391*n^5 + 52472779*n^4 - 140788713*n^3 + 210641238*n^2 - 166531044*n + 54330068)*a(n) = (n-1)^2*(109683470*n^10 - 1564397770*n^9 + 9692963299*n^8 - 34227043418*n^7 + 75994068609*n^6 - 110509975758*n^5 + 106422212572*n^4 - 67092633284*n^3 + 26619112256*n^2 - 6034674112*n + 596279040)*a(n-1) - (1736373702*n^12 - 31711114890*n^11 + 261518988565*n^10 - 1286766506127*n^9 + 4203065855621*n^8 - 9590033857111*n^7 + 15649936441072*n^6 - 18370855225904*n^5 + 15360506258964*n^4 - 8896962441876*n^3 + 3377234408016*n^2 - 751582555104*n + 73915071552)*a(n-2) - (n-2)^2*(36279917*n^10 - 590014481*n^9 + 4216923435*n^8 - 17398379754*n^7 + 45760527058*n^6 - 79915647314*n^5 + 93501944898*n^4 - 72055169071*n^3 + 34824212688*n^2 - 9481092472*n + 1100757336)*a(n-3) - (n-2)^2*(843719*n^6 - 5284077*n^5 + 13396609*n^4 - 17487127*n^3 + 12303648*n^2 - 4393232*n + 621656)*(n-3)^4*a(n-4). - Vaclav Kotesovec, Nov 04 2012

MATHEMATICA

Table[Sum[Binomial[n, k]^2*Binomial[n+k, k]^3, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 04 2012 *)

PROG

(PARI) a(n) = sum(k=0, n, binomial(n, k)^2*binomial(n+k, k)^3); \\ Michel Marcus, Mar 10 2016

CROSSREFS

Cf. A218693, A112019, A014178, A014180, A218689, A218692.

Sequence in context: A124070 A203763 A041619 * A289196 A266907 A288547

Adjacent sequences:  A111965 A111966 A111967 * A111969 A111970 A111971

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 28 2005

STATUS

approved

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Last modified June 16 19:43 EDT 2019. Contains 324155 sequences. (Running on oeis4.)