A258659 E.g.f. A(x) satisfies: A(x) = exp( Integral A(x)^(1/2) * Integral 1/A(x)^(3/2) dx dx ).
1, 1, 3, 18, 189, 3024, 68607, 2095632, 82908441, 4124203776, 251944606683, 18542621357568, 1618221395188629, 165230649971380224, 19514714407120367127, 2639737292796971845632, 405452689572115086887601, 70178277847381981514366976, 13596354857453497541480646963, 2930800480466007704630652960768
Offset: 0
Keywords
Examples
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! +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..240
Programs
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Mathematica
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 *)
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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)} for(n=0,20,print1(a(2*n),", ")) -
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)} for(n=0,20,print1(a(2*n),", ")) -
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 */
Formula
E.g.f. A(x) satisfies: A(x) = exp( Integral 1/A(x)^(1/2) * Integral A(x)^(3/2) dx dx ).
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
a(2*n) = A261000(n). - Michael Somos, Jun 17 2017
Comments