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A188946

Binomial partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

1, 7, 103, 1969, 41935, 947737, 22248409, 536310271, 13183283743, 328970388985, 8307368234473, 211822788505951, 5444571611722369, 140892128574440887, 3667015053678269095, 95918056089104563489, 2519845343307697266943

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..16.

FORMULA

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*binomial(3*k,k).

Recurrence: (n+3)^2*a(n+3)-(30*n^2+150*n+187)*a(n+2)+57*(n+2)^2*a(n+1)-28*(n+1)*(n+2)*a(n)=0.

E.g.f.: exp(x)*F(1/3,2/3;1,1;27*x), where F(a1,a2;b1;z) is a hypergeometric series.

a(n) = hypergeom([1/3, 2/3, -n], [1, 1], -27). - Vladimir Reshetnikov, Oct 15 2017

a(n) ~ 2^(2*n+1) * 7^(n+1) / (3^(5/2)*Pi*n). - Vaclav Kotesovec, Nov 27 2017

MATHEMATICA

Table[Sum[Binomial[n, k]Binomial[2k, k]Binomial[3k, k], {k, 0, n}], {n, 0, 16}]

Table[HypergeometricPFQ[{1/3, 2/3, -n}, {1, 1}, -27], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2017 *)

PROG

(Maxima) makelist(sum(binomial(n, k)*binomial(2*k, k)*binomial(3*k, k), k, 0, n), n, 0, 16);

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(2*k, k)*binomial(3*k, k)); \\ Michel Marcus, Oct 15 2017

(Magma) [&+[Binomial(n, k)*Binomial(2*k, k)*Binomial(3*k, k): k in [0..n]]: n in [0.. 18]]; // Vincenzo Librandi, Oct 16 2017

CROSSREFS

Cf. A006480, A188441, A188918, A152297.

Sequence in context: A246914 A384950 A274670 * A368440 A365014 A234292

Adjacent sequences: A188943 A188944 A188945 * A188947 A188948 A188949

KEYWORD

nonn,easy

AUTHOR

Emanuele Munarini, Apr 14 2011

STATUS

approved

URL: https://oeis.org/A188946

⇱ A188946 - OEIS