%I #9 Apr 30 2014 01:33:24
%S 1,1,3,13,81,669,6955,88505,1346209,23998521,493956467,11596542533,
%T 307301505073,9110471500693,299893197116059,10888674034993905,
%U 433549376981078593,18833037527449398129,888439543634687700579
%N O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*(k+1)*x).
%F a(n+1) = row sums of A071951. - _Michael Somos_, Feb 25 2012
%F G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-(k+1)*(k+2)*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2013
%e O.g.f.: A(x) = 1 + x/(1-2x) + x^2/((1-2x)*(1-6x)) + x^3/((1-2x)*(1-6x)*(1-12x)) + x^4/((1-2x)*(1-6x)*(1-12x)*(1-20x)) + ...
%e Also generated by iterated binomial transforms in the following way:
%e [1,3,13,81,669,6955,88505,...] = BINOMIAL^2([1,1,5,31,253,2673,34833,..]);
%e [1,5,31,253,2673,34833,541879,...] = BINOMIAL^4([1,1,7,57,577,7389,...]);
%e [1,7,57,577,7389,115983,2151493,...] = BINOMIAL^6([1,1,9,91,1101,16497,...]);
%e [1,9,91,1101,16497,301669,..] = BINOMIAL^8([1,1,11,133,1873,32061,..]);
%e [1,11,133,1873,32061,666579,...] = BINOMIAL^10([1,1,13,183,2941,56529,...]);
%e etc.
%o (PARI) a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-j*(j+1)*x+x*O(x^n))), n)
%o (PARI) {a(n) = sum( k=0, n, sum( i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! ))} /* _Michael Somos_, Feb 25 2012 */
%Y Cf. A071951, A135920, A124373.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 06 2007