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%I #3 Feb 01 2013 22:03:10
%S 1,3,10,39,183,1026,6695,49623,411050,3763599,37757055,411894882,
%T 4854301087,61459583007,831926801290,11989221944871,183273754945959,
%U 2961997167865410,50462267599637975,903853088211536295,16980055625062979306,333846342195447641343
%N G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (3 + k*x)/(1 + 3*k*x + k^2*x^2).
%e G.f.: A(x) = 1 + 3*x + 10*x^2 + 39*x^3 + 183*x^4 + 1026*x^5 + 6695*x^6 +...
%e where
%e A(x) = 1 + x*(3+x)/(1+3*x+x^2) + 2!*x^2*(3+x)*(3+2*x)/((1+3*x+x^2)*(1+6*x+4*x^2)) + 3!*x^3*(3+x)*(3+2*x)*(3+3*x)/((1+3*x+x^2)*(1+6*x+4*x^2)*(1+9*x+9*x^2)) + 4!*x^4*(3+x)*(3+2*x)*(3+3*x)*(3+4*x)/((1+3*x+x^2)*(1+6*x+4*x^2)*(1+9*x+9*x^2)*(1+12*x+16*x^2)) +...
%o (PARI) {a(n)=polcoeff( sum(m=0, n, m!*x^m*prod(k=1, m, (3+k*x)/(1+3*k*x+k^2*x^2 +x*O(x^n))) ), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A208237, A136127.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 01 2013
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