A346665 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(5*k,k) / (4*k + 1).
1, 0, 4, 22, 172, 1409, 12216, 109904, 1016876, 9614584, 92490261, 902364918, 8907507708, 88802649446, 892833960460, 9042639746819, 92171773008828, 944819352291920, 9733592874215112, 100725697334689896, 1046535959932600141, 10913073121311627481, 114175868855824821752
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..958
Programs
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Maple
A346665 := proc(n) add((-1)^(n-k)*binomial(n,k)*binomial(5*k,k)/(4*k+1),k=0..n) ; end proc: seq(A346665(n),n=0..80); # R. J. Mathar, Aug 17 2023
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Mathematica
Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 22}] nmax = 22; 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 = 22; CoefficientList[Series[Sum[(Binomial[5 k, k]/(4 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x] Table[(-1)^n HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5, -n}, {1/2, 3/4, 1, 5/4}, 3125/256], {n, 0, 22}] -
PARI
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*binomial(5*k,k)/(4*k + 1)); \\ Michel Marcus, Jul 28 2021
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) ~ 2869^(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) -(n-1) *(1845*n^3 -1333*n^2 -238*n +240)*a(n-1) -4*(n-1) *(2485*n^3 -7263*n^2 +7388*n -2580) *a(n-2) -2*(n-1) *(n-2) *(8095*n^2 -24029*n +18924) *a(n-3) -4*(n-1) *(n-2) *(n-3) *(2805*n -5578) *a(n-4) -2869*(n-1) *(n-2) *(n-3) *(n-4) *a(n-5)=0. - R. J. Mathar, Aug 17 2023
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