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A262002 L.g.f.: log( Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2 + 1) ).

%I A262002 #9 Sep 08 2015 18:16:35
%S A262002 2,6,28,164,1132,8916,78608,765904,8170752,94755776,1187551088,
%T A262002 16004096144,230910861872,3553052000336,58100442762368,
%U A262002 1006457051734784,18415695160624192,354980774078690496,7190981550797464448,152744987909458781824,3395058926880381635712,78814259879097446800256
%N A262002 L.g.f.: log( Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2 + 1) ).
%F A262002 O.g.f.: 1 - 1/G(x) where G(x) is the g.f. of A262001.
%e A262002 L.g.f.: L(x) = 2*x + 6*x^2/2 + 28*x^3/3 + 164*x^4/4 + 1132*x^5/5 + 8916*x^6/6 + 78608*x^7/7 + 765904*x^8/8 + 8170752*x^9/9 + 94755776*x^10/10 +...
%e A262002 such that
%e A262002 exp(L(x)) = 1 + 2*x + 10*x^2/2! + 100*x^3/3! + 1700*x^4/4! + 44200*x^5/5! + 1635400*x^6/6! +...+ A101686(n)*x^n/n! +...
%e A262002 where A101686(n) = Product_{k=1..n} (k^2+1).
%e A262002 Also, given the o.g.f. A(x) = Sum_{n>=1} a(n)*x^n,
%e A262002 o.g.f.: A(x) = 2*x + 6*x^2 + 28*x^3 + 164*x^4 + 1132*x^5 + 8916*x^6 +...
%e A262002 then
%e A262002 1/(1 - A(x)) = 1 + 2*x + 10*x^2 + 60*x^3 + 400*x^4 + 2900*x^5 + 22700*x^6 + 191600*x^7 + 1746400*x^8 + 17230000*x^9 + 184348000*x^10 +...+ A262001(n)*x^n +...
%o A262002 (PARI) {a(n) = n*polcoeff( log(sum(m=0,n+1,x^m/m!*prod(k=1,m,k^2+1)) +x*O(x^n)), n)}
%o A262002 for(n=1,30,print1(a(n),", "))
%Y A262002 Cf. A101686, A262001.
%K A262002 nonn
%O A262002 1,1
%A A262002 _Paul D. Hanna_, Sep 08 2015