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%I #13 Nov 03 2014 02:49:24
%S 1,2,11,96,1117,16226,282503,5732736,132852169,3461582882,
%T 100170007859,3187366835040,110607201645061,4157078373725762,
%U 168223595493553871,7292428324945552896,337148168449629548497,16559308764450704877506,861070129828668874777883
%N G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (2*k - x) / (1 - 2*k*x).
%C Compare to a g.f. of Pell numbers (A000129):
%C Sum_{n>=0} x^n * Product_{k=1..n} (2*k + x)/(1 + 2*k*x) = 1/(1-2*x-x^2).
%C Limit n->infinity A231229(n) / A231277(n) = 2. - _Vaclav Kotesovec_, Nov 02 2014
%H Vaclav Kotesovec, <a href="/A231229/b231229.txt">Table of n, a(n) for n = 0..282</a>
%F Self-convolution square-root yields A231277.
%F a(n) == (n+1) (mod 4).
%F a(n) ~ 2^(n-1) * n! / (log(2))^(n+1). - _Vaclav Kotesovec_, Nov 02 2014
%e G.f.: A(x) = 1 + 2*x + 11*x^2 + 96*x^3 + 1117*x^4 + 16226*x^5 + 282503*x^6 +...
%e where
%e A(x) = 1 + x*(2-x)/(1-2*x) + x^2*(2-x)*(4-x)/((1-2*x)*(1-4*x)) + x^3*(2-x)*(4-x)*(6-x)/((1-2*x)*(1-4*x)*(1-6*x)) + x^4*(2-x)*(4-x)*(6-x)*(8-x)/((1-2*x)*(1-4*x)*(1-6*x)*(1-8*x)) +...
%e The g.f. equals the square of an integer series (A231277):
%e A(x)^(1/2) = 1 + x + 5*x^2 + 43*x^3 + 503*x^4 + 7395*x^5 + 130417*x^6 + 2677347*x^7 + 62652163*x^8 + 1645424927*x^9 + 47918249503*x^10 +...
%e because the g.f. A(x) is congruent to 1/(1-x)^2 modulo 4.
%o (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, (2*k-x)/(1-2*k*x +x*O(x^n)))), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A231172, A231277.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 05 2013
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