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A258657 E.g.f. A(x) satisfies: A(x) = exp( Integral A(x)^2 * Integral 1/A(x)^6 dx dx ).

%I A258657 #13 Mar 17 2024 10:26:42
%S A258657 1,1,3,63,1449,79569,4933467,538446447,64308790161,11702872619361,
%T A258657 2287996769790963,625623383080392543,181555646645074399929,
%U A258657 69630517380932205676209,28096571683140898776853707,14391876463069615780007165967,7705942233644896318623038389281,5082509306223148713110677899062721
%N A258657 E.g.f. A(x) satisfies: A(x) = exp( Integral A(x)^2 * Integral 1/A(x)^6 dx dx ).
%C A258657 More generally, we have the identity for real t:
%C A258657 * if G(x) = exp( Integral G(x)^t * Integral 1/G(x)^(3*t) dx dx ),
%C A258657 then G(x) = exp( Integral 1/G(x)^t * Integral G(x)^(3*t) dx dx ).
%H A258657 Vaclav Kotesovec, <a href="/A258657/b258657.txt">Table of n, a(n) for n = 0..190</a>
%F A258657 E.g.f. A(x) satisfies: A(x) = exp( Integral 1/A(x)^2 * Integral A(x)^6 dx dx ).
%F A258657 a(n) ~ c * 2^(7*n+1) * Pi^(n+1) * n^(2*n) / (exp(2*n) * Gamma(1/4)^(4*n)), where c = 2^(7/4) / (Gamma(1/4) * Pi^(3/4)) = 0.393150663417355411930715... . - _Vaclav Kotesovec_, Jun 15 2015, updated Mar 17 2024
%e A258657 E.g.f. A(x) = 1 + x^2/2! + 3*x^4/4! + 63*x^6/6! + 1449*x^8/8! + 79569*x^10/10! +...
%t A258657 a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Sqrt @ JacobiDC[ x, -1], {x, 0, m}]]]; (* _Michael Somos_, Jun 17 2016 *)
%o A258657 (PARI) {a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A^2 * intformal(1/A^6 + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
%o A258657 for(n=0,20,print1(a(2*n),", "))
%o A258657 (PARI) {a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( 1/A^2 * intformal(A^6 + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
%o A258657 for(n=0,20,print1(a(2*n),", "))
%Y A258657 Cf. A159600, A258658, A258659.
%K A258657 nonn
%O A258657 0,3
%A A258657 _Paul D. Hanna_, Jun 06 2015