A372201 -id:A372201 - cp's OEIS frontend

A372200 E.g.f. A(x) satisfies A(x) = exp( 2 * x / (1 - x * A(x)^(1/2))^2 ).

1, 2, 12, 116, 1600, 28832, 643864, 17190392, 534707296, 19003345568, 760054943464, 33798503960168, 1654577248619728, 88437537019736816, 5125378381513865752, 320163561707158120568, 21445740148760729672896, 1533498858453023915309888

Offset: 0

Seiichi Manyama, Apr 21 2024

nonn

Cf. A161630, A372201.

Cf. A161635, A372160.

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

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A161635.

If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.

A372203 E.g.f. A(x) satisfies A(x) = exp( 3 * x * (1 + x * A(x)^(1/3))^3 ).

Original entry on oeis.org

1, 3, 27, 297, 4581, 87363, 2014389, 54516969, 1695624345, 59673787587, 2345478318369, 101896766246817, 4850500185441909, 251143864572078819, 14055460408215741069, 845667848072862801657, 54441943452534058086321, 3734566046400701428294275

Offset: 0

Seiichi Manyama, Apr 21 2024

nonn

Cf. A365030, A372201.

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

E.g.f.: A(x) = B(x)^3 where B(x) is the e.g.f. of A365030.

If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(s*k,n-k)/k!.

cp's OEIS Frontend

A372200 E.g.f. A(x) satisfies A(x) = exp( 2 * x / (1 - x * A(x)^(1/2))^2 ).

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