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O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k*(k+1)/2 + x) / (1 + k*(k+1)/2*x).
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%I #8 Oct 28 2014 18:08:51

%S 1,1,3,10,51,370,3691,48525,812089,16832928,422860609,12649706416,

%T 444120983433,18078156682309,844323149201499,44838127594166770,

%U 2686250544297734323,180295858504407010026,13473490672899749784979,1114874245392058455432873

%N O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k*(k+1)/2 + x) / (1 + k*(k+1)/2*x).

%C Compare to an o.g.f. of Genocchi numbers of the second kind (A000366):

%C Sum_{n>=0} x^n * Product_{k=1..n} k*(k+1)/2 / (1 + k*(k+1)/2*x).

%H Vaclav Kotesovec, <a href="/A230740/b230740.txt">Table of n, a(n) for n = 0..260</a>

%F a(n) ~ 2^(n+6) * n^(2*n+7/2) / (exp(2*n) * Pi^(2*n+5/2)). - _Vaclav Kotesovec_, Oct 28 2014

%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 51*x^4 + 370*x^5 + 3691*x^6 +...

%e where

%e A(x) = 1 + x*(1+x)/(1+x) + x^2*(1+x)*(3+x)/((1+x)*(1+3*x)) + x^3*(1+x)*(3+x)*(6+x)/((1+x)*(1+3*x)*(1+6*x)) + x^4*(1+x)*(3+x)*(6+x)*(10+x)/((1+x)*(1+3*x)*(1+6*x)*(1+10*x)) +...

%o (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, k*(k+1)/2+x+x*O(x^n))/prod(k=1, m, 1+k*(k+1)/2*x+x*O(x^n))), n)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A230682.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 28 2013