A381443 -id:A381443 - cp's OEIS frontend

A381450 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cosh(x))^3 ).

1, 3, 24, 339, 7056, 195855, 6819840, 286105071, 14055420288, 791783681499, 50327779368960, 3563709848656683, 278223968271034368, 23744747385054558759, 2199369837961901789184, 219748696455778150645575, 23559108001707680103628800, 2697737574531326391439989171

Offset: 0

Seiichi Manyama, Feb 23 2025

nonn

As stated in the comment of A185951, A185951(n,0) = 0^n.

Index entries for reversions of series

Cf. A381171, A381449.
Cf. A377554, A381443.
Cf. A185951, A381448.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(3*n+3, k)*a185951(n, k))/(n+1);

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cosh(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A381448.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(3*n+3,k) * A185951(n,k).

A381430 E.g.f. A(x) satisfies A(x) = 1 + sinh(x*A(x)^3).

Original entry on oeis.org

1, 1, 6, 73, 1368, 34861, 1126368, 44135701, 2034072960, 107823563641, 6463383851520, 432331180935457, 31924171503581184, 2579483385868484005, 226383845487041421312, 21445302563389991287981, 2180974075392495296544768, 237009522316557393020262001, 27409082977094100068471537664

Offset: 0

Seiichi Manyama, Feb 23 2025

nonn

Index entries for reversions of series

Cf. A162653, A198865.

Cf. A136630, A381443.

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(3*n+1, k)*a136630(n, k))/(3*n+1);

E.g.f.: ( (1/x) * Series_Reversion( x/(1 + sinh(x))^3 ) )^(1/3).

a(n) = (1/(3*n+1)) * Sum_{k=0..n} k! * binomial(3*n+1,k) * A136630(n,k).

cp's OEIS Frontend

A381450 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cosh(x))^3 ).

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