A216357 Expansion of 1/( (1-16*x)*(1+4*x)^2 )^(1/4).

1, 2, 38, 404, 5510, 74492, 1048924, 15004776, 217943238, 3200089580, 47405806708, 707305846936, 10616181542044, 160142807848792, 2426097698458360, 36890818642990544, 562772826273060678, 8609639617006367052, 132048790603779592196, 2029851945081220214200

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..830

Formula

G.f.: exp(Sum_{n>=1} A070775(n)*x^n/n) where A070775(n) = Sum_{k=0..n} binomial(4*n,4*k).

a(n) ~ GAMMA(3/4) * 2^(4*n+1/2) / (Pi* sqrt(5) * n^(3/4)). - Vaclav Kotesovec, Jul 31 2014

a(n) = ((64*n-80)*a(n-2)+(12*n-10)*a(n-1))/n. - Robert Israel, Dec 09 2016

Example

G.f.: A(x) = 1 + 2*x + 38*x^2 + 404*x^3 + 5510*x^4 + 74492*x^5 + 1048924*x^6 + ...
where 1/A(x)^4 = 1 - 8*x - 112*x^2 - 256*x^3.
The logarithm of the g.f. begins:
log(A(x)) = x + 2*x^2/2 + 72*x^3/3 + 992*x^4/4 + 16512*x^5/5 + 261632*x^6/6 + 4196352*x^7/7 + ... + A070775(n)*x^n/n + ...

Maple

f:= gfun:-rectoproc({(48+64*n)*a(n)+(14+12*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 2}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Dec 09 2016

Mathematica

a = DifferenceRoot[Function[{a, n}, {(48+64n) a[n] + (14+12n) a[1+n] + (-2-n) a[2+n] == 0, a[0] == 1, a[1] == 2}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Sep 16 2022, after Robert Israel *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(4*m, 4*j))*x^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Crossrefs

Cf. A070775, A216316, A216358.

Sequence in context: A303618 A179503 A126731 * A046845 A098772 A207320

Adjacent sequences: A216354 A216355 A216356 * A216358 A216359 A216360

Keyword

nonn

Author

Paul D. Hanna, Sep 04 2012

Status

approved