A205543 Logarithmic derivative of the Bell numbers (A000110).

1, 3, 10, 39, 171, 822, 4271, 23759, 140518, 878883, 5789015, 40019058, 289513303, 2186421919, 17199606090, 140662816543, 1193865048363, 10499107480518, 95528651305671, 898071593401559, 8712429618413678, 87118795125708283, 896925422648691735

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: A350915 A221585 A083862 * A137590 A352174 A124532

Adjacent sequences: A205540 A205541 A205542 * A205544 A205545 A205546

Keyword

nonn

Author

Paul D. Hanna, Jan 28 2012

Status

approved