A206397 E.g.f. A(x) = series reversion of log(1+x)-x^3/3.

1, 1, 3, 21, 171, 1821, 24123, 373941, 6693291, 135897741, 3081969243, 77250233061, 2120715880011, 63277499072061, 2039050439495163, 70571948084252181, 2610905715855178731, 102824333281385113581

Offset

1,3

Links

Table of n, a(n) for n=1..18.

Formula

a(n)=(sum(k=0..n-1, (n+k-1)!*sum(j=0..k, (-1)^j/(k-j)!*sum(i=0..min(j,(n+j-1)/3),(1/i!)*(-1)^i* stirling1(n-3*i+j-1,j-i)/(3^i*(n-3*i+j-1)!))))), n>0.

a(n) ~ n^(n-1) / (sqrt(s*(2+s^3)) * exp(n) * r^(n-1/2)), where s = 1/3*(-1 + (25/2 - (3*sqrt(69))/2)^(1/3) + (1/2*(25 + 3*sqrt(69)))^(1/3)) = 0.75487766624669276... is the root of the equation s^2*(1+s) = 1 and r = log(1+s) - s^3/3 = 0.4190125789786... - Vaclav Kotesovec, Jan 22 2014

Mathematica

Rest[CoefficientList[InverseSeries[Series[Log[1+x]-x^3/3, {x, 0, 20}], x], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 22 2014 *)

Prog

(Maxima) a(n):=(sum((n+k-1)!*sum((-1)^j/(k-j)!*sum((1/i!)*(-1)^i*stirling1(n-3*i+j-1, j-i)/(3^i*(n-3*i+j-1)!), i, 0, min(j, (n+j-1)/3)), j, 0, k), k, 0, n-1));

Crossrefs

Sequence in context: A372089 A206178 A233861 * A247480 A365136 A228923

Adjacent sequences: A206394 A206395 A206396 * A206398 A206399 A206400

Keyword

nonn

Author

Vladimir Kruchinin, Feb 07 2012

Status

approved