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

%I A346647 #17 Aug 17 2023 04:50:44
%S A346647 1,2,8,54,460,4361,43988,462580,5014252,55624944,628432101,7205500484,
%T A346647 83632219892,980710882430,11601345881748,138278231052451,
%U A346647 1659037424218780,20020306637339944,242835190201382648,2958961154058610552,36203518795424475661
%N A346647 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*k,k) / (4*k + 1).
%C A346647 Binomial transform of A002294.
%H A346647 Seiichi Manyama, <a href="/A346647/b346647.txt">Table of n, a(n) for n = 0..500</a>
%F A346647 G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^3 * A(x)^5.
%F A346647 G.f.: Sum_{k>=0} ( binomial(5*k,k) / (4*k + 1) ) * x^k / (1 - x)^(k+1).
%F A346647 a(n) ~ 3381^(n + 3/2) / (78125 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - _Vaclav Kotesovec_, Jul 30 2021
%F A346647 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
%p A346647 A346647 := proc(n)
%p A346647     hypergeom([-n,1/5,2/5,3/5,4/5],[1/2,3/4,1,5/4],-3125/256) ;
%p A346647     simplify(%) ;
%p A346647 end proc:
%p A346647 seq(A346647(n),n=0..40) ; # _R. J. Mathar_, Jan 10 2023
%t A346647 Table[Sum[Binomial[n, k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
%t A346647 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]
%t A346647 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]
%t A346647 Table[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5, -n}, {1/2, 3/4, 1, 5/4}, -3125/256], {n, 0, 20}]
%o A346647 (PARI) a(n) = sum(k=0, n, binomial(n,k)*binomial(5*k,k)/(4*k + 1)); \\ _Michel Marcus_, Jul 26 2021
%Y A346647 Cf. A002294, A007317, A188687, A227035, A346646, A346648, A346649, A346650.
%K A346647 nonn
%O A346647 0,2
%A A346647 _Ilya Gutkovskiy_, Jul 26 2021