A382059 -id:A382059 - cp's OEIS frontend

A380515 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A380513, A380514, A380516.
Cf. A080893, A380511.
Cf. A091695, A250917, A380512.
Cf. A002293, A006632, A370057, A382059.

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

a(n) = 3 * n! * Sum_{k=0..n-1} binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-4*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^3 ) ). - Seiichi Manyama, Mar 15 2025

A382058 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 67, 1465, 44541, 1735681, 82527439, 4632741905, 299875704697, 21989097804961, 1801520077445331, 163092373817762137, 16168084561101716725, 1741946677697976052577, 202668693570279026375671, 25324088113475137179021601, 3382305512670022948599733233, 480858973986045019386825360577

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A161629, A382059.
Cf. A161635, A382032.
Cf. A001764, A377546, A382033.

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

Let F(x) be the e.g.f. of A377546. F(x) = log(A(x))/x = B(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^2 ) ).
a(n) = 2 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.

cp's OEIS Frontend

A380515 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A382058 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A380515 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A382058 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A380515 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

A382058 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.