A228411 G.f.: ( (1 - sqrt(1-32*x)) / (16*x) )^(1/4).

1, 2, 26, 476, 10150, 236060, 5807076, 148581048, 3913759878, 105424703020, 2890693930124, 80413849328904, 2263896023453532, 64381391412987672, 1846729385267277960, 53367451809002583408, 1552274439636853988550, 45408989873571191613900, 1335107241077282661195900

Offset: 0

Author

Paul D. Hanna, Nov 09 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 26*x^2 + 476*x^3 + 10150*x^4 + 236060*x^5 +...
where
A(x)^4 = 1 + 8*x + 128*x^2 + 2560*x^3 + 57344*x^4 + 1376256*x^5 +...+ A000108(n)*8^n*x^n +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..665

Crossrefs

Cf. A000108, A159318.

Programs

Mathematica

CoefficientList[Series[((1-Sqrt[1-32*x])/(16*x))^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 10 2013 *)
Table[8^n Binomial[2 n + 1/4, n]/(8 n + 1), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 12 2016 *)

PARI

/* G.f.: ( (1 - sqrt(1-32*x)) / (16*x) )^(1/4): */
{a(n)=polcoeff(( (1 - sqrt(1-32*x +x^2*O(x^n))) / (16*x) )^(1/4),n)}
for(n=0,30,print1(a(n),", "))

PARI

/* G.f.: A(x) = C(8*x)^(1/4), C(x) is Catalan function: */
{a(n)=polcoeff((serreverse(x-8*x^2 +x^2*O(x^n))/x)^(1/4),n)}
for(n=0,30,print1(a(n),", "))

PARI

/* G.f.: A(x) = exp( x*A(x)^8 + Integral(A(x)^8 dx) ): */
{a(n)=local(A=1+x);for(i=1,n,A=exp(x*A^8+intformal(A^8+x*O(x^n))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = exp( x*A(x)^8 + Integral(A(x)^8 dx) ).

(2) A(x)^4 = 1 + 8*x*A(x)^8, thus A(x) = C(8*x)^(1/4) where C(x) is the Catalan function (A000108).

a(n) ~ 2^(5*n-3+1/4)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 10 2013

D-finite with recurrence: n*(4*n+1)*a(n) -2*(8*n-3)*(8*n-7)*a(n-1)=0. - R. J. Mathar, Oct 08 2016

a(n) = 8^n*binomial(2*n + 1/4, n)/(8*n + 1). - Vladimir Reshetnikov, Oct 12 2016