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%I #3 Aug 30 2014 21:08:02
%S 1,1,7,51,425,3879,36527,355333,3531175,35673875,365179885,3777991337,
%T 39430009247,414567124053,4386228722281,46659584847835,
%U 498701253293129,5352318710976505,57655365854918487,623105208980304843,6753999316026236871,73403038257774972101,799674458063926645329
%N G.f.: sqrt( Sum_{n>=0} x^n / (1-x)^(4*n+1) * [Sum_{k=0..2*n} C(2*n,k)^2 * x^k]^2 ).
%C Self-convolution equals A246570.
%F G.f.: A(x) = 1 + x + 7*x^2 + 51*x^3 + 425*x^4 + 3879*x^5 + 36527*x^6 +...
%F such that
%F A(x)^2 = 1/(1-x) + x/(1-x)^5 * (1 + 2^2*x + x^2)^2
%F + x^2/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2
%F + x^3/(1-x)^13 * (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^2 +...
%F Explicitly,
%F A(x)^2 = 1 + 2*x + 15*x^2 + 116*x^3 + 1001*x^4 + 9322*x^5 + 89363*x^6 +...+ A246570(n)*x^n +...
%o (PARI) /* By definition: */
%o {a(n)=local(A=1); A = sqrt( sum(m=0, n, x^m/(1-x)^(4*m+1) * sum(k=0, 2*m, binomial(2*m, k)^2 * x^k)^2 +x*O(x^n)) ); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A246563, A246570, A246571, A246573.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 30 2014
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