A185310 - OEIS

%I #38 Nov 03 2014 03:48:24

%S 1,1,2,7,29,154,961,6989,57856,537507,5539661,62713770,773755901,

%T 10333514049,148523400880,2285980425751,37513428757945,

%U 653836562581682,12062552192649377,234841212544184453,4811561492994497360,103490537790094623515,2331542163236964185653

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

%H Vaclav Kotesovec, <a href="/A185310/b185310.txt">Table of n, a(n) for n = 0..250</a>

%F a(n) ~ c * d^n * n!, where d = 1.022144551255634378856..., c = 1.3357568735203273768... . - _Vaclav Kotesovec_, Nov 02 2014

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 154*x^5 + 961*x^6 + 6989*x^7 +...

%e where

%e A(x) = 1 + (1+x)*x/(1+x+x^2) + 2!*(1+2*x)^2*x^2/((1+x+2*x^2)*(1+2*x+4*x^2)) + 3!*(1+3*x)^3*x^3/((1+x+3*x^2)*(1+2*x+6*x^2)*(1+3*x+9*x^2)) + 4!*(1+4*x)^4*x^4/((1+x+4*x^2)*(1+2*x+8*x^2)*(1+3*x+12*x^2)*(1+4*x+16*x^2)) +...

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

%o for(n=0, 25, print1(a(n), ", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 01 2013