A258970 - OEIS

%I #16 Jun 06 2024 12:25:30

%S 1,2,8,80,1088,19328,422912,10987520,330555392,11300913152,

%T 432717037568,18344259092480,852932666851328,43157160112160768,

%U 2360748463307620352,138821061188696145920,8732741520836633034752,585172975239737913638912,41612642758392039581155328

%N E.g.f.: A'(x) = 1 + A(x)^4, with A(0)=1.

%H Vaclav Kotesovec, <a href="/A258970/b258970.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) ~ n! * d^(n+1/3) / (3^(1/3) * GAMMA(1/3) * n^(2/3)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(4*j-1) = 4*sqrt(2)/(Pi + log(3-2*sqrt(2))) = 4.10260201986929...

%F E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^4) dx ). - _Paul D. Hanna_, Jun 16 2015

%e A(x) = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1088*x^4/4! + 19328*x^5/5! + ...

%e A'(x) = 2 + 8*x + 40*x^2 + 544*x^3/3 + 2416*x^4/3 + 52864*x^5/15 + ...

%e 1 + A(x)^4 = 2 + 8*x + 40*x^2 + 544*x^3/3 + 2416*x^4/3 + 52864*x^5/15 + ...

%t nmax=20; Subscript[a,0]=1; egf=Sum[Subscript[a,k]*x^k, {k,0,nmax+1}]; Table[Subscript[a,k]*k!, {k,0,nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^4-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]

%o (PARI) {a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^4 +x*O(x^n)) )); n!*polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jun 16 2015

%Y Cf. A000831, A258969, A258971, A258994, A258901.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jun 15 2015