A235374 - OEIS

%I #10 Jan 29 2014 06:47:21

%S 1,1,2,14,88,1096,11792,209744,3211648,74050816,1474533632,

%T 41710490624,1023774788608,34285617473536,1001167232079872,

%U 38715438665007104,1311494550010298368,57488503079879213056,2217017970860729434112,108599775372146808848384

%N E.g.f. satisfies: A'(x) = A(x)^6 * A(-x)^4 with A(0) = 1.

%H Vaclav Kotesovec, <a href="/A235374/b235374.txt">Table of n, a(n) for n = 0..390</a>

%H Vaclav Kotesovec, <a href="/A235374/a235374.txt">Recurrence (of order 8)</a>

%F E.g.f.: 1/(1 - Series_Reversion( Integral (1-x^2)^4 dx )).

%F a(n) ~ n! * 2^(4/5) * (315/128)^(n+1/5) / (Gamma(1/5) * 5^(1/5) * n^(4/5)). - _Vaclav Kotesovec_, Jan 29 2014

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 88*x^4/4! + 1096*x^5/5! +...

%e Related series.

%e A(x)^6 = 1 + 6*x + 42*x^2/2! + 384*x^3/3! + 4368*x^4/4! + 60096*x^5/5! +...

%e Note that 1 - 1/A(x) is an odd function:

%e 1 - 1/A(x) = x + 8*x^3/3! + 496*x^5/5! + 81728*x^7/7! +...

%e where Series_Reversion(1 - 1/A(x)) = Integral (1-x^2)^4 dx.

%t CoefficientList[1/(1 - InverseSeries[Series[Integrate[(1-x^2)^4,x],{x,0,20}],x]),x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 28 2014 *)

%o (PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^6*subst(A, x, -x)^4 +x*O(x^n) )); n!*polcoeff(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n)=local(A=1); A=1/(1-serreverse(intformal((1-x^2 +x*O(x^n))^4))); n!*polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A235370, A235371, A235372, A235373.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 07 2014