A346647 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*k,k) / (4*k + 1).

1, 2, 8, 54, 460, 4361, 43988, 462580, 5014252, 55624944, 628432101, 7205500484, 83632219892, 980710882430, 11601345881748, 138278231052451, 1659037424218780, 20020306637339944, 242835190201382648, 2958961154058610552, 36203518795424475661

Offset

0,2

Comments

Binomial transform of A002294.

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)^3 * A(x)^5.

G.f.: Sum_{k>=0} ( binomial(5*k,k) / (4*k + 1) ) * x^k / (1 - x)^(k+1).

a(n) ~ 3381^(n + 3/2) / (78125 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 30 2021

D-finite with recurrence +8*n*(4*n+1) *(2*n-1)*(4*n-1)*a(n) +(-4405*n^4 +9322*n^3 -7655*n^2 +2978*n -480)*a(n-1) +12*(n-1) *(1255*n^3 -3829*n^2 +4204*n -1640) *a(n-2) -2*(n-1) *(n-2) *(10655*n^2 -32221*n +26076) *a(n-3) +4*(n-1) *(n-2) *(n-3)*(3445*n -6922) *a(n-4) -3381*(n-1)*(n-2) *(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 17 2023

Maple

A346647 := proc(n)
    hypergeom([-n, 1/5, 2/5, 3/5, 4/5], [1/2, 3/4, 1, 5/4], -3125/256) ;
    simplify(%) ;
end proc:
seq(A346647(n), n=0..40) ; # R. J. Mathar, Jan 10 2023

Mathematica

Table[Sum[Binomial[n, k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^3 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 20; CoefficientList[Series[Sum[(Binomial[5 k, k]/(4 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5, -n}, {1/2, 3/4, 1, 5/4}, -3125/256], {n, 0, 20}]

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(5*k, k)/(4*k + 1)); \\ Michel Marcus, Jul 26 2021

Crossrefs

Cf. A002294, A007317, A188687, A227035, A346646, A346648, A346649, A346650.

Sequence in context: A079503 A052694 A069729 * A354690 A052599 A352648

Adjacent sequences: A346644 A346645 A346646 * A346648 A346649 A346650

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 26 2021

Status

approved