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A201731 a(n) = [x^n/n!] log( (n - (n-1)*exp(x)) / (n+1 - n*exp(x)) ). 2
1, 4, 54, 1544, 75750, 5676492, 603041334, 86210654224, 15958892198070, 3713676157320020, 1061084890984465446, 365202873520507832856, 149027843082185351506950, 71144948740332156241755868, 39282974873393643411310747350, 24840594864924259316810487005216 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..16.

FORMULA

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

EXAMPLE

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.

PROG

(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)}

CROSSREFS

Cf. A000629, A201338, A201732.

Sequence in context: A111161 A216733 A203039 * A225823 A265004 A284747

Adjacent sequences: A201728 A201729 A201730 * A201732 A201733 A201734

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 04 2011

STATUS

approved

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Last modified December 9 11:21 EST 2022. Contains 358700 sequences. (Running on oeis4.)