A380640 -id:A380640 - cp's OEIS frontend

A380639 Expansion of e.g.f. exp(x/(1 - 2*x)^2).

1, 1, 9, 97, 1297, 20961, 398041, 8678209, 213337377, 5830560577, 175187949481, 5734893998241, 203021979225649, 7724154592735777, 314158263983430777, 13597375157683820161, 623802598335834369601, 30228101725367033318529, 1542430410234859308052297

Offset: 0

Seiichi Manyama, Jan 28 2025

Cf. A380636, A380640.

Cf. A082579.

a(n) = n!*sum(k=0, n, 2^k*binomial(2*n-k-1, k)/(n-k)!);

a(n) = n! * Sum_{k=0..n} 2^k * binomial(2*n-k-1,k)/(n-k)!.
E.g.f.: exp( Sum_{k>=1} k * 2^(k-1) * x^k ).
a(n) ~ 2^n * n^(n - 1/6) / (sqrt(3) * exp(n - 3*n^(2/3)/2 + 1/24)). - Vaclav Kotesovec, Jan 29 2025

A380643 Expansion of e.g.f. exp(xG(3x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 19, 865, 63289, 6402421, 827951491, 130454402149, 24246255965905, 5193341198368489, 1259626725043888051, 341256073037890028041, 102138911537774675080969, 33470594059698797005874845, 11918817613356955871120346979, 4582850483720783516657005897741

Offset: 0

Seiichi Manyama, Jan 29 2025

nonn

Cf. A002293, A380637.

Cf. A080893, A380640.

a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, 3^k*binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));

a(n) = 3 * n! * Sum_{k=0..n-1} 3^k * binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
E.g.f. A(x) satisfies x = log(A(x)) * (1 - 3*log(A(x)))^3.
a(n) = 3^(n-1)*U(1-n, 2-4*n, 1/3), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025
E.g.f.: exp( Series_Reversion( x*(1-3*x)^3 ) ). - Seiichi Manyama, Mar 16 2025

cp's OEIS Frontend

A380639 Expansion of e.g.f. exp(x/(1 - 2*x)^2).

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A380643 Expansion of e.g.f. exp(xG(3x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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cp's OEIS Frontend

A380639 Expansion of e.g.f. exp(x/(1 - 2*x)^2).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A380643 Expansion of e.g.f. exp(x*G(3*x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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PARI

Formula

A380643 Expansion of e.g.f. exp(xG(3x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.