A370463 E.g.f. satisfies A(x) = log(1 + x)/(1 - A(x))^3.

0, 1, 5, 74, 1704, 54474, 2225394, 110786976, 6506273544, 440368208280, 33752787590136, 2889747086330400, 273333159994125984, 28307010099549881088, 3185660442523728449664, 387117483236717961052800, 50518567433159392237036416, 7046383438320021239186859264

Offset

0,3

Comments

In general, if k >=1 and e.g.f. satisfies A(x) = log(1 + x)/(1 - A(x))^k, then a(n) ~ (k+1)^(k/2 - 1) * n^(n-1) / (k^((k-1)/2) * exp(n + k^k/(2*(k+1)^(k+1))) * (exp(k^k/(k+1)^(k+1)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Jan 24 2026

Links

Table of n, a(n) for n=0..17.

Index entries for reversions of series

Formula

a(n) = Sum_{k=1..n} (4*k-2)!/(3*k-1)! * Stirling1(n,k).

E.g.f.: Series_Reversion( exp(x * (1 - x)^3) - 1 ). - Seiichi Manyama, Sep 09 2024

a(n) ~ 2 * n^(n-1) / (3 * exp(n + 27/512) * (exp(27/256) - 1)^(n - 1/2)). - Vaclav Kotesovec, Jan 24 2026

Mathematica

Table[Sum[(4*k - 2)!/(3*k - 1)! * StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2026 *)

Prog

(PARI) a(n) = sum(k=1, n, (4*k-2)!/(3*k-1)!*stirling(n, k, 1));

Crossrefs

Cf. A087138, A370462.

Sequence in context: A065894 A233013 A383678 * A126740 A251666 A192564

Adjacent sequences: A370460 A370461 A370462 * A370464 A370465 A370466

Keyword

nonn

Author

Seiichi Manyama, Mar 18 2024

Status

approved