A080893 -id:A080893 - cp's OEIS frontend

A375175 Expansion of e.g.f. exp( (exp( (exp(4*x) - 1)/2 ) - 1)/2 ).

1, 1, 7, 63, 713, 9753, 156111, 2858103, 58845105, 1344371793, 33713484151, 919838859151, 27105053988793, 857310780134825, 28953291147179007, 1039373409620929671, 39505610599553955809, 1584411299793530257697, 66846625774893448843879

Offset: 0

Seiichi Manyama, Aug 02 2024

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A080893, A375173.

Cf. A004211.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((exp((exp(4*x)-1)/2)-1)/2)))

a(n) = Sum_{k=0..n} 4^(n-k) * Stirling2(n,k) * A004211(k) = 4^n * Sum_{k=0..n} (1/2)^k * Stirling2(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

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

A375175 Expansion of e.g.f. exp( (exp( (exp(4*x) - 1)/2 ) - 1)/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

A375175 Expansion of e.g.f. exp( (exp( (exp(4*x) - 1)/2 ) - 1)/2 ).

Views

Author

Keywords

Links

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.

Views

Author

Keywords

Crossrefs

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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.