A236957 - OEIS

%I #4 Apr 09 2014 21:03:20

%S 1,1,3,31,297,5521,90843,2421391,56778897,1965992161,59991229683,

%T 2551838332351,96020199171897,4840069070838001,216719978024072523,

%U 12622971840715547311,655783794933664894497,43320949673000323765441,2562378473386758135272163,189242342019412261693784671

%N E.g.f. satisfies: A'(x) = A(x)^7 * A(-x)^4 with A(0) = 1.

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

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 297*x^4/4! + 5521*x^5/5! +...

%e Related series.

%e A(x)^7 = 1 + 7*x + 63*x^2/2! + 805*x^3/3! + 13041*x^4/4! + 261247*x^5/5! +...

%e Note that 1 - 1/A(x)^2 is an odd function:

%e 1 - 1/A(x)^2 = 2*x + 32*x^3/3! + 4352*x^5/5! + 1605632*x^7/7! +...

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

%o (PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^7*subst(A^4, x, -x) +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/(1-2*serreverse(intformal((1-4*x^2 +x*O(x^n))^(4/2))))^(1/2); n!*polcoeff(A, n)}

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

%Y  Cf. A236953, A236954, A236955, A236956, A235374, A236958.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 09 2014