cp's OEIS Frontend

A258659 E.g.f. A(x) satisfies: A(x) = exp( Integral A(x)^(1/2) * Integral 1/A(x)^(3/2) dx dx ).

%I A258659 #16 Mar 16 2024 13:01:12
%S A258659 1,1,3,18,189,3024,68607,2095632,82908441,4124203776,251944606683,
%T A258659 18542621357568,1618221395188629,165230649971380224,
%U A258659 19514714407120367127,2639737292796971845632,405452689572115086887601,70178277847381981514366976,13596354857453497541480646963,2930800480466007704630652960768
%N A258659 E.g.f. A(x) satisfies: A(x) = exp( Integral A(x)^(1/2) * Integral 1/A(x)^(3/2) dx dx ).
%C A258659 More generally, we have the identity for real t:
%C A258659 * if G(x) = exp( Integral G(x)^t * Integral 1/G(x)^(3*t) dx dx ),
%C A258659 then G(x) = exp( Integral 1/G(x)^t * Integral G(x)^(3*t) dx dx ).
%H A258659 Vaclav Kotesovec, <a href="/A258659/b258659.txt">Table of n, a(n) for n = 0..240</a>
%F A258659 E.g.f. A(x) satisfies: A(x) = exp( Integral 1/A(x)^(1/2) * Integral A(x)^(3/2) dx dx ).
%F A258659 a(n) ~ c * d^n * n!^2 * sqrt(n), where d = 32*Pi / Gamma(1/4)^4 = 0.58180245681734198604520486465..., c = 1.31298754327535054303509412... . - _Vaclav Kotesovec_, Jun 15 2015, updated Mar 16 2024
%F A258659 a(2*n) = A261000(n). - _Michael Somos_, Jun 17 2017
%e A258659 E.g.f. A(x) = 1 + x^2/2! + 3*x^4/4! + 18*x^6/6! + 189*x^8/8! + 3024*x^10/10! +...
%t A258659 nmax = 40; A = 1; Do[A = Exp[Integrate[A^(1/2)*Integrate[1/A^(3/2), x], x] + O[x]^nmax], nmax]; CoefficientList[A, x^2]*Range[0, nmax-2, 2]! (* _Jean-François Alcover_, Nov 27 2017 *)
%o A258659 (PARI) {a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A^(1/2) * intformal(1/A^(3/2) + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
%o A258659 for(n=0,20,print1(a(2*n),", "))
%o A258659 (PARI) {a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( 1/A^(1/2) * intformal(A^(3/2) + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
%o A258659 for(n=0,20,print1(a(2*n),", "))
%o A258659 (PARI) {a(n) = if( n<0, 0, my(m = 2*n); m! * polcoeff( exp( intformal( serreverse( intformal( 1 / sqrt(1 + x^4/4 + x * O(x^m)) ) ) ) ), m))}; /* _Michael Somos_, Jun 17 2017 */
%Y A258659 Cf. A159600, A258657, A258658, A261000.
%K A258659 nonn
%O A258659 0,3
%A A258659 _Paul D. Hanna_, Jun 06 2015