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A205543 Logarithmic derivative of the Bell numbers (A000110). 2
1, 3, 10, 39, 171, 822, 4271, 23759, 140518, 878883, 5789015, 40019058, 289513303, 2186421919, 17199606090, 140662816543, 1193865048363, 10499107480518, 95528651305671, 898071593401559, 8712429618413678, 87118795125708283, 896925422648691735 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = number of indecomposable partitions (A074664) of [n+3] in which n+3 lies in a doubleton block (see Link). - David Callan, Oct 08 2014

LINKS

Table of n, a(n) for n=1..23.

David Callan, A combinatorial interpretation for this sequence

FORMULA

L.g.f.: log( Sum_{n>=0} x^n / Product_{k=1..n} (1 - k*x) ).

EXAMPLE

L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 39*x^4/4 + 171*x^5/5 + 822*x^6/6 +...

where exponentiation yields the o.g.f. of the Bell numbers:

exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 203*x^6 + 877*x^7 +...

which equals the series:

exp(L(x)) = 1 + x/(1-x) + x^2/((1-x)*(1-2*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)) +...

PROG

(PARI) {a(n)=n*polcoeff(log(sum(m=0, n, x^m/prod(k=1, m, 1-k*x +x*O(x^n)))), n)}

CROSSREFS

Cf. A000110.

Sequence in context: A245378 A221585 A083862 * A137590 A124532 A074728

Adjacent sequences:  A205540 A205541 A205542 * A205544 A205545 A205546

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 28 2012

STATUS

approved

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Last modified December 7 08:21 EST 2016. Contains 278849 sequences.