A365030 -id:A365030 - cp's OEIS frontend

A363744 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x))^2).

1, 1, 5, 31, 313, 3981, 63841, 1223419, 27378737, 701091001, 20221662241, 649032795951, 22945630163017, 886151307346501, 37121193546044609, 1676607954371120611, 81222976991097364321, 4201418329450141471473, 231127287514383805458625

Offset: 0

Seiichi Manyama, Aug 17 2023

nonn

Cf. A125500, A365030.

Cf. A161635.

Join[{1}, Table[n! * Sum[(n-k+1)^(k-1) * Binomial[2*k,n-k]/k!, {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)

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

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

a(n) ~ sqrt((1 + r*s)*(1 + 3*r*s) / (2*(1 + 2*r + 4*r^2*s + 2*r^3*s^2))) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.302307732979052080722256232095444259577495... and s = 2.910394288602135748195482733301939282588478379746... are real roots of the system of equations exp(r*(1 + r*s)^2) = s, 2*s*r^2*(1 + r*s) = 1. - Vaclav Kotesovec, Nov 18 2023

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

A363744 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x))^2).

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