%I #7 Oct 01 2015 17:31:18
%S 1,1,1,1,7,31,91,211,997,5941,27181,97021,369907,1809211,9180991,
%T 40941031,170062027,753752971,3645183691,17473250251,79444369189,
%U 356738879533,1662097580353,7957682872873,37696688946691,175245959453491,816849622436251,3873243058472971,18507865654295389
%N G.f.: [ Sum_{n>=0} (4*n)!/(n!)^4 * x^(4*n)/(1-x)^(4*n+4) ]^(1/4).
%F G.f. satisfies: A(x) = 1/(1-x) * Sum_{n>=0} A262013(n) * (x*A(x))^(4*n).
%F G.f.: A(x) = (1/x) * Series_Reversion( x / (G(x^4) + x) ) where G(x) is the g.f. of A262013.
%e G.f.: A(x) = 1 + x + x^2 + x^3 + 7*x^4 + 31*x^5 + 91*x^6 + 211*x^7 +...
%e such that
%e A(x)^4 = 1/(1-x)^4 + 24*x^4/(1-x)^8 + 2520*x^8/(1-x)^12 + 369600*x^12/(1-x)^16 + 63063000*x^16/(1-x)^20 + 11732745024*x^20/(1-x)^24 +...+ (4*n)!/(n!)^4*x^(4*n)/(1-x)^(4*n+4) +...
%e explicitly,
%e A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 59*x^4 + 248*x^5 + 948*x^6 + 3000*x^7 + 10605*x^8 + 49468*x^9 + 238030*x^10 +...
%e Also, we have the series
%e x/Series_Reversion(x*A(x)) = 1+x + 6*x^4 + 432*x^8 + 45960*x^12 + 5780034*x^16 + 797957244*x^20 + 116916528960*x^24 + 17852845828752*x^28 + 2810058672255120*x^32 + 452703723158137776*x^36 + 74282858140993920000*x^40 +...+ A262013(n)*x^(4*n) +...
%e so that
%e A(x)*(1-x) = 1 + 6*x^4*A(x)^4 + 432*x^8*A(x)^8 + 45960*x^12*A(x)^12 + 5780034*x^16*A(x)^16 +...+ A262013(n)*(x*A(x))^(4*n) +...
%o (PARI) {a(n)=polcoeff(sum(m=0, n,x^(4*m)/(1-x +x*O(x^n))^(4*m+4)*(4*m)!/(m!)^4)^(1/4), n)}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A262013.
%K nonn
%O 0,5
%A _Paul D. Hanna_, Sep 11 2015
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