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

1, 1, 3, 63, 1449, 79569, 4933467, 538446447, 64308790161, 11702872619361, 2287996769790963, 625623383080392543, 181555646645074399929, 69630517380932205676209, 28096571683140898776853707, 14391876463069615780007165967, 7705942233644896318623038389281, 5082509306223148713110677899062721

Offset: 0

Author

Paul D. Hanna, Jun 06 2015

Keywords

• nonn

Comments

More generally, we have the identity for real t:

* if G(x) = exp( Integral G(x)^t * Integral 1/G(x)^(3*t) dx dx ),

then G(x) = exp( Integral 1/G(x)^t * Integral G(x)^(3*t) dx dx ).

Examples

			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! +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..190

Crossrefs

Cf. A159600, A258658, A258659.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Sqrt @ JacobiDC[ x, -1], {x, 0, m}]]]; (* Michael Somos, Jun 17 2016 *)

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)}
for(n=0,20,print1(a(2*n),", "))

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)}
for(n=0,20,print1(a(2*n),", "))

Formula

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

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