cp's OEIS Frontend

A205543 Logarithmic derivative of the Bell numbers (A000110).

%I A205543 #12 Oct 08 2014 17:01:31
%S A205543 1,3,10,39,171,822,4271,23759,140518,878883,5789015,40019058,
%T A205543 289513303,2186421919,17199606090,140662816543,1193865048363,
%U A205543 10499107480518,95528651305671,898071593401559,8712429618413678,87118795125708283,896925422648691735
%N A205543 Logarithmic derivative of the Bell numbers (A000110).
%C A205543 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
%H A205543 David Callan, <a href="/A205543/a205543.pdf">A combinatorial interpretation for this sequence</a>
%F A205543 L.g.f.: log( Sum_{n>=0} x^n / Product_{k=1..n} (1 - k*x) ).
%e A205543 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 +...
%e A205543 where exponentiation yields the o.g.f. of the Bell numbers:
%e A205543 exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 203*x^6 + 877*x^7 +...
%e A205543 which equals the series:
%e A205543 exp(L(x)) = 1 + x/(1-x) + x^2/((1-x)*(1-2*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)) +...
%o A205543 (PARI) {a(n)=n*polcoeff(log(sum(m=0,n, x^m/prod(k=1,m, 1-k*x +x*O(x^n)))),n)}
%Y A205543 Cf. A000110.
%K A205543 nonn
%O A205543 1,2
%A A205543 _Paul D. Hanna_, Jan 28 2012