A381172 -id:A381172 - cp's OEIS frontend

A381171 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + x*cosh(x)) ).

1, 1, 2, 9, 72, 725, 8640, 124117, 2117248, 41477193, 913305600, 22371549761, 604476094464, 17858943664861, 572524035586048, 19793963392789965, 734249332747960320, 29090332675789113617, 1225991945551031304192, 54765451909152748484857, 2584803582762012599910400

Offset: 0

Seiichi Manyama, Feb 16 2025

nonn

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

Index entries for reversions of series

Cf. A162654, A215364, A381172.
Cf. A162653, A381173, A381181.
Cf. A185951.

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

E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x * A(x)) ).

a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(n+1,k) * A185951(n,k).

A381175 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cos(x * A(x)) ).

Original entry on oeis.org

1, 1, 6, 69, 1224, 29465, 898320, 33187133, 1441200768, 71956238769, 4061414246400, 255737764687669, 17773804761259008, 1351494159065894857, 111608708333568036864, 9947544079380663728685, 951770403836914402099200, 97301151510219112917218657, 10585077723403580668983902208

Offset: 0

Seiichi Manyama, Feb 16 2025

nonn

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

Cf. A381173, A381174, A381176.
Cf. A364984, A381172.
Cf. A185951.

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

a(n) = Sum_{k=0..n} k! * binomial(n+2*k+1,k)/(n+2*k+1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

cp's OEIS Frontend

A381171 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + x*cosh(x)) ).

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