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%I #16 Oct 01 2015 17:30:50
%S 1,6,432,45960,5780034,797957244,116916528960,17852845828752,
%T 2810058672255120,452703723158137776,74282858140993920000,
%U 12371608762947252317376,2085965999078265151837416,355369363809372393287259600,61077516024443872565938037760,10577549099569850669961138879936,1844022794086994489463720605589954,323353325204017719895246887849230460
%N G.f. A(x) satisfies: A(x)^4 = Sum_{n>=0} (4*n)!/(n!)^4 * x^n / A(x)^(4*n).
%F G.f. A(x) satisfies:
%F (1) A(x)^4 = x / Series_Reversion( Sum_{n>=0} (4*n)!/(n!)^4 * x^(n+1) ).
%F (2) Series_Reversion( x/A(x)^4 ) = Sum_{n>=0} (4*n)!/(n!)^4 * x^(n+1).
%F (3) A(x^4) = -x + x / Series_Reversion( x*F(x) ) where F(x)^4 = Sum_{n>=0} (4*n)!/(n!)^4 * x^(4*n)/(1-x)^(4*n+4).
%F (4) (1/x) * Series_Reversion( x / (A(x^4) + x) ) equals the g.f. of A262012.
%e G.f.: A(x) = 1 + 6*x + 432*x^2 + 45960*x^3 + 5780034*x^4 + 797957244*x^5 +...
%e such that
%e A(x)^4 = 1 + 24*x/A(x)^4 + 2520*x^2/A(x)^8 + 369600*x^3/A(x)^12 + 63063000*x^4/A(x)^16 + 11732745024*x^5/A(x)^20 +...+ (4*n)!/(n!)^4 * x^n/A(x)^(4*n) +...
%e where
%e A(x)^4 = 1 + 24*x + 1944*x^2 + 215808*x^3 + 27736920*x^4 + 3879912960*x^5 + 573515224128*x^6 + 88128590118912*x^7 +...+ A262010(n)*x^n +...
%e Also,
%e (1/x)*Series_Reversion( x/(A(x^4) + x) ) = 1 + x + x^2 + x^3 + 7*x^4 + 31*x^5 + 91*x^6 + 211*x^7 + 997*x^8 + 5941*x^9 + 27181*x^10 + 97021*x^11 + 369907*x^12 +...+ A262012(n)*x^n +...
%o (PARI) {a(n) = local(A); A = ( x/serreverse( x*sum(m=0, n, (4*m)!/(m!)^4*x^m +x*O(x^n)) ) )^(1/4) ; polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = local(A); A = -x + x/serreverse(x*sum(m=0, n, x^(4*m)/(1-x +O(x^(4*n+2)))^(4*m+4)*(4*m)!/(m!)^4)^(1/4)) ; polcoeff(A, 4*n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A262010, A262012.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 11 2015
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