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%I #17 Aug 17 2023 04:45:54
%S 1,0,4,22,172,1409,12216,109904,1016876,9614584,92490261,902364918,
%T 8907507708,88802649446,892833960460,9042639746819,92171773008828,
%U 944819352291920,9733592874215112,100725697334689896,1046535959932600141,10913073121311627481,114175868855824821752
%N a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(5*k,k) / (4*k + 1).
%C Inverse binomial transform of A002294.
%H Seiichi Manyama, <a href="/A346665/b346665.txt">Table of n, a(n) for n = 0..958</a>
%F G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^3 * A(x)^5.
%F G.f.: Sum_{k>=0} ( binomial(5*k,k) / (4*k + 1) ) * x^k / (1 + x)^(k+1).
%F a(n) ~ 2869^(n + 3/2) / (78125 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - _Vaclav Kotesovec_, Jul 30 2021
%F 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
%p A346665 := proc(n)
%p add((-1)^(n-k)*binomial(n,k)*binomial(5*k,k)/(4*k+1),k=0..n) ;
%p end proc:
%p seq(A346665(n),n=0..80); # _R. J. Mathar_, Aug 17 2023
%t Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 22}]
%t 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]
%t 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]
%t 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}]
%o (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
%Y Cf. A002294, A005043, A346628, A346647, A346664, A346666, A346667, A346668.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 27 2021
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