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G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + (2*k+1)*x).
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%I #11 Nov 01 2014 04:20:15

%S 1,1,2,5,13,40,127,443,1610,6207,24919,104440,453913,2042537,9488242,

%T 45403797,223433077,1128619968,5844484375,30981783123,167949231882,

%U 929951967519,5254958379951,30276315741008,177717403802001,1061989593129105,6456342053713346

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

%H Vaclav Kotesovec, <a href="/A202153/b202153.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: 1 + x*(1+x)/( G(0) - x*(1+x) ) where G(k) = 1 + x + x^2*(2*k+1)- x*(2*x*k+1+3*x)/G(k+1) ; G(0) = x*(1+x) if k->infinity; (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Feb 04 2013

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 40*x^5 + 127*x^6 +...

%e where

%e A(x) = 1 + x*(1+x) + x^2*(1+x)*(1+3*x) + x^3*(1+x)*(1+3*x)*(1+5*x) +...

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

%Y Cf. A124380.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 13 2011