A152297 -id:A152297 - cp's OEIS frontend

A188918 Alternate partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

1, 5, 85, 1595, 33055, 723701, 16429435, 382643525, 9082868245, 218790563255, 5332206228085, 131194789234955, 3253536973286245, 81224561099580155, 2039348104811147845, 51455631680563483835, 1303889832725451598495

Offset: 0

Emanuele Munarini, Apr 14 2011

Cf. A006480, A188441, A188946, A152297.

Table[Sum[(-1)^(n-k)*Binomial[2k,k]Binomial[3k,k],{k,0,n}],{n,0,16}] (* fixed by Vaclav Kotesovec, Nov 27 2017 *)

makelist(sum(binomial(2*k,k)*binomial(3*k,k)*(-1)^(n-k),k,0,n),n,0,16);

a(n) = sum((-1)^(n-k)*binomial(2*k,k)*binomial(3*k,k),k=0..n).
Recurrence: (n+2)^2*a(n+2)-(26*n^2+77*n+56)*a(n+1)-3*(9*n^2+27*n+20)*a(n)=0.
G.f.: F(1/3,2/3;1;27*x)/(1+x), where F(a1,a2;b1;z) is a hypergeometric series.
a(n) ~ 3^(3*n + 7/2) / (56*Pi*n). - Vaclav Kotesovec, Nov 27 2017

A188946 Binomial partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

Original entry on oeis.org

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

Offset: 0

Emanuele Munarini, Apr 14 2011

Cf. A006480, A188441, A188918, A152297.

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

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 *)

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

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

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

cp's OEIS Frontend

A188918 Alternate partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

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