A387086 -id:A387086 - cp's OEIS frontend

A385719 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.

1, 4, 38, 428, 5204, 66104, 863840, 11515308, 155779966, 2131436392, 29426804398, 409254436452, 5726378247412, 80535621269208, 1137609359823936, 16130112288879248, 229462608491483364, 3273749607191060480, 46826932120849617128, 671341041479214814160, 9644654058165119642624

Offset: 0

Seiichi Manyama, Aug 17 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..850

Cf. A183161, A208977, A387084, A387086.

Cf. A002295, A004355, A385497.

nmax = 20; CoefficientList[Series[Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] - 1)/3], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)

Sum_{k=0..n} a(k) * a(n-k) = A385497(n).
G.f.: 1/sqrt(1 - 4*x*g^4*(3-g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: g/sqrt((2-g) * (6-5*g)) where g = 1+x*g^6 is the g.f. of A002295.
a(n) ~ 2^(6*n - 1/2) * 3^(6*n + 3/4) / (Gamma(1/4) * n^(3/4) * 5^(5*n + 1/4)) * (1 + 7*Gamma(1/4)^2/(48*Pi*sqrt(30*n))). - Vaclav Kotesovec, Aug 20 2025

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

Original entry on oeis.org

1, 3, 23, 211, 2095, 21752, 232439, 2534182, 28041295, 313833025, 3544160216, 40318629754, 461455158383, 5308453068900, 61333295856750, 711305543582150, 8276351877367663, 96576953297406377, 1129842469637643485, 13248082583624602575, 155660344852055352760

Offset: 0

Seiichi Manyama, Aug 16 2025

nonn

Cf. A183161, A208977, A387086.

Cf. A001449, A002294, A079589.

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 *)

Sum_{k=0..n} a(k) * a(n-k) = A079589(n).
G.f.: 1/sqrt(1 - x*g^3*(5+g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: g/sqrt(5-4*g) where g = 1+x*g^5 is the g.f. of A002294.
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
a(n) ~ 5^(5*n + 3/4) / (Gamma(1/4) * n^(3/4) * 2^(8*n + 7/4)). - Vaclav Kotesovec, Aug 20 2025

cp's OEIS Frontend

A385719 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula