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A387084 Expansion of B(x)/sqrt(1 + 4*(B(x)-1)/5), where B(x) is the g.f. of A001449.

%I A387084 #19 Aug 20 2025 07:03:48
%S A387084 1,3,23,211,2095,21752,232439,2534182,28041295,313833025,3544160216,
%T A387084 40318629754,461455158383,5308453068900,61333295856750,
%U A387084 711305543582150,8276351877367663,96576953297406377,1129842469637643485,13248082583624602575,155660344852055352760
%N A387084 Expansion of B(x)/sqrt(1 + 4*(B(x)-1)/5), where B(x) is the g.f. of A001449.
%F A387084 Sum_{k=0..n} a(k) * a(n-k) = A079589(n).
%F A387084 G.f.: 1/sqrt(1 - x*g^3*(5+g)) where g = 1+x*g^5 is the g.f. of A002294.
%F A387084 G.f.: g/sqrt(5-4*g) where g = 1+x*g^5 is the g.f. of A002294.
%F A387084 Conjecture D-finite with recurrence 3902464*n*(8*n-5) *(8*n-3)*(8*n-1) *(8*n+1)*a(n) +80*(-12565760000*n^5 +68448000000*n^4 -163457516000*n^3 +200475354000*n^2 -122843089511*n +29804717943)*a(n-1) +125000*(134055000*n^5 -1109795000*n^4 +3726971625*n^3 -6307124125*n^2 +5325821766*n -1769460798)*a(n-2) +48828125*(-1556875*n^5 +15845625*n^4 -60659875*n^3 +103818375*n^2 -67764178*n +1391424)*a(n-3) -152587890625 *(5*n-16)*(n-3) *(5*n-19)*(5*n-18) *(5*n-17)*a(n-4)=0. - _R. J. Mathar_, Aug 19 2025
%F A387084 a(n) ~ 5^(5*n + 3/4) / (Gamma(1/4) * n^(3/4) * 2^(8*n + 7/4)). - _Vaclav Kotesovec_, Aug 20 2025
%t A387084 nmax = 25; CoefficientList[Series[Sum[Binomial[5*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 4*(Sum[Binomial[5*n, n]*x^n, {n, 0, nmax}] - 1)/5], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 20 2025 *)
%Y A387084 Cf. A183161, A208977, A387086.
%Y A387084 Cf. A001449, A002294, A079589.
%K A387084 nonn
%O A387084 0,2
%A A387084 _Seiichi Manyama_, Aug 16 2025