A381450 -id:A381450 - cp's OEIS frontend

A381449 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cosh(x))^2 ).

1, 2, 10, 90, 1224, 22450, 517920, 14395514, 468414464, 17474840226, 735559614720, 34491849224602, 1783268816102400, 100786369113730898, 6182264844496971776, 409065938149354422330, 29043282491002728284160, 2202461172795524123296834, 177675452451923238962528256

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, A381450.

Cf. A185951, A381447.

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(2*n+2, k)*a185951(n, k))/(n+1);

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cosh(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381447.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * A185951(n,k).

A381443 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + sinh(x))^3 ).

Original entry on oeis.org

1, 3, 24, 333, 6720, 179523, 5992800, 240498261, 11287790592, 607019415075, 36813049552896, 2486167829854173, 185070328813031424, 15056826823777670883, 1329283990371617820672, 126573877370649849898149, 12930948581449447912243200, 1410875453109072905123881923

Offset: 0

Seiichi Manyama, Feb 23 2025

nonn

Index entries for reversions of series

Cf. A377554, A381450.

Cf. A136630, A381430.

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+3, k)*a136630(n, k))/(n+1);

E.g.f. A(x) satisfies A(x) = (1 + sinh(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A381430.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(3*n+3,k) * A136630(n,k).

cp's OEIS Frontend

A381449 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cosh(x))^2 ).

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