A380675 -id:A380675 - cp's OEIS frontend

A380674 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x * (1 - x)^2) ).

1, 3, 25, 370, 8097, 237096, 8733601, 388380000, 20253654945, 1212334652800, 81937521020841, 6172429566120192, 512850795552978625, 46594245206418954240, 4595466275857015549425, 488993161791784338804736, 55839856392986843905585089, 6811561624203525171739852800

Offset: 0

Seiichi Manyama, Jan 30 2025

nonn

Index entries for reversions of series

Cf. A377832, A380665, A380666, A380675.

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

E.g.f. A(x) satisfies A(x) = exp(x * A(x) * (1 - x*A(x))^2)/(1 - x*A(x))^2.

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(3*n-3*k+1,n-k)/k!.

A380779 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x / (1 + x)) / (1 + x)^2 ).

Original entry on oeis.org

1, 3, 23, 298, 5529, 134496, 4062631, 146903184, 6193969137, 298577002240, 16204658051031, 978156957629952, 65017249611283657, 4719532271850590208, 371519503997940966375, 31526820740816885549056, 2869134152226896957509089, 278763390556764407051452416

Offset: 0

Seiichi Manyama, Feb 02 2025

nonn

Index entries for reversions of series

Cf. A377829, A380778, A380780, A380781.

Cf. A380675.

a(n, q=1, r=1, s=1, t=-1, u=2) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial(r*u*n+((s-r)*u+t)*k+q*u, n-k)/k!);

E.g.f. A(x) satisfies A(x) = exp( x * A(x) / (1 + x*A(x)) ) * (1 + x*A(x))^2.

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n-k+2,n-k)/k!.

cp's OEIS Frontend

A380674 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-x * (1 - x)^2) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula