A346646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) / (3*k + 1).

1, 2, 7, 38, 257, 1935, 15505, 129519, 1115061, 9823160, 88121887, 802227794, 7392428009, 68819554003, 646276497617, 6114880542117, 58237420303109, 557850829527246, 5370956411708779, 51947475492561014, 504492516832543885, 4917564488572565160

Offset

0,2

Comments

Binomial transform of A002293.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^2 * A(x)^4.

G.f.: Sum_{k>=0} ( binomial(4*k,k) / (3*k + 1) ) * x^k / (1 - x)^(k+1).

a(n) ~ 283^(n + 3/2) / (2048 * sqrt(2*Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) -2*(2*n-1) *(91*n^2 -91*n +24)*a(n-1) +6*(n-1) *(155*n^2 -310*n +167)*a(n-2) -438*(n-1) *(n-2)*(2*n-3) *a(n-3) +283*(n-1)*(n-2) *(n-3)*a(n-4)=0. - R. J. Mathar, Aug 17 2023

Maple

A346646 := proc(n)
    hypergeom([-n, 1/4, 1/2, 3/4], [2/3, 1, 4/3], -256/27) ;
    simplify(%) ;
end proc:
seq(A346646(n), n=0..40) ; # R. J. Mathar, Jan 10 2023

Mathematica

Table[Sum[Binomial[n, k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 21; CoefficientList[Series[Sum[(Binomial[4 k, k]/(3 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {2/3, 1, 4/3}, -256/27], {n, 0, 21}]

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(4*k, k)/(3*k + 1)); \\ Michel Marcus, Jul 26 2021

Crossrefs

Cf. A002293, A007317, A104859, A188687, A226974, A346647, A346648, A346649, A346650.

Sequence in context: A256032 A209006 A168492 * A032109 A368232 A337026

Adjacent sequences: A346643 A346644 A346645 * A346647 A346648 A346649

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 26 2021

Status

approved