A201731 a(n) = [x^n/n!] log( (n - (n-1)*exp(x)) / (n+1 - n*exp(x)) ).

1, 4, 54, 1544, 75750, 5676492, 603041334, 86210654224, 15958892198070, 3713676157320020, 1061084890984465446, 365202873520507832856, 149027843082185351506950, 71144948740332156241755868, 39282974873393643411310747350, 24840594864924259316810487005216

Offset: 1

Author

Paul D. Hanna, Dec 04 2011

Keywords

• nonn

Comments

The function log((n - (n-1)*exp(x))/(n+1 - n*exp(x))) equals the n-th iteration of log(1/(2-exp(x))), the e.g.f. of A000629 (with offset 1), where A000629(n) is the number of necklaces of partitions of n+1 labeled beads.

Examples

			Let G(x) = log(1/(2-exp(x))) then the coefficients of x^n/n! in the k-th iteration of G(x) begin:
k=1: [(1), 2, 6, 26, 150, 1082, 9366, 94586, ..., A000629(n-1), ...];
k=2: [1,(4), 24, 196, 2040, 25924, 390264, 6804676, ..., A201338(n), ...];
k=3: [1, 6,(54), 654, 9990, 184686, 4015494, 100531374, ...];
k=4: [1, 8, 96, (1544), 31200, 760328, 21721056, 712459784, ...];
k=5: [1, 10, 150, 3010, (75750), 2295010, 81378150, 3307983010, ...];
k=6: [1, 12, 216, 5196, 156600, (5676492), 240593976, 11679764556, ...];
k=7: [1, 14, 294, 8246, 289590, 12224534, (603041334), 34053651926, ...];
k=8: [1, 16, 384, 12304, 493440, 23777296, 1338417024, (86210654224), ...]; ...
where the coefficients in parenthesis form the initial terms of this sequence.
		

Crossrefs

Cf. A000629, A201338, A201732.

Programs

PARI

{a(n)=n!*polcoeff(log((n - (n-1)*exp(x+x*O(x^n)))/(n+1 - n*exp(x+x*O(x^n)))),n)}

Formula

a(n) = (n+1) * A201732(n+1).