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A262001 G.f.: 1/(1 - x*F'(x)/F(x)) where F(x) = Sum_{n>=0} x^n/n!*Product_{k=1..n} (k^2 + 1).

%I A262001 #15 Sep 10 2015 12:37:42
%S A262001 1,2,10,60,400,2900,22700,191600,1746400,17230000,184348000,
%T A262001 2140118000,26925784000,366118706000,5359236310000,84077608400000,
%U A262001 1407341155720000,25027454132360000,471046698018440000,9351091483806800000,195213433887227200000,4274234604872786800000,97924306054031604400000
%N A262001 G.f.: 1/(1 - x*F'(x)/F(x)) where F(x) = Sum_{n>=0} x^n/n!*Product_{k=1..n} (k^2 + 1).
%C A262001 Cf. A262002, which is defined by: Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2+1) = exp( Sum_{n>=1} A262002(n)*x^n/n ).
%C A262001 Sum of all terms results in the 10-adic number x = ...5211383820350605156083728207423149062180073.
%F A262001 G.f.: 1/(1 - G(x)) where G(x) is an o.g.f. of A262002.
%F A262001 a(n) == 0 (mod 10) for n>1.
%e A262001 O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 400*x^4 + 2900*x^5 + 22700*x^6 +...
%e A262001 where
%e A262001 1 - 1/A(x) =  2*x + 6*x^2 + 28*x^3 + 164*x^4 + 1132*x^5 + 8916*x^6 + 78608*x^7 + 765904*x^8 + 8170752*x^9 +...+ A262002(n)*x^n +...
%e A262001 Note that if we define the logarithmic series:
%e A262001 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 +...+ A262002(n)*x^n/n +...
%e A262001 then 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! +... where A101686(n) = Product_{k=1..n} (k^2+1).
%o A262001 (PARI) {a(n) = local(A=1,L=log(sum(m=0,n+1,x^m/m!*prod(k=1,m,k^2+1)) +x*O(x^n))); A=1/(1 - x*L'); polcoeff(A +x*O(x^n), n)}
%o A262001 for(n=0,30,print1(a(n),", "))
%Y A262001 Cf. A262002.
%K A262001 nonn
%O A262001 0,2
%A A262001 _Paul D. Hanna_, Sep 08 2015