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
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..500
Programs
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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
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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}] -
PARI
a(n) = sum(k=0, n, binomial(n,k)*binomial(4*k,k)/(3*k + 1)); \\ Michel Marcus, Jul 26 2021
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
Comments