OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Nachum Dershowitz, Touchard’s Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2 arXiv:1110.6638 [math.NT], 2011 (the sequence b-hat(n) defined at the end of Section 5.1.1).
FORMULA
a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*b(k)^2, where b() = A126869().
E.g.f.: (exp(x) * I_0(2*x))^2 = square of e.g.f. of A002426. - Michael Somos, Jun 27 2012
From Mark van Hoeij, May 07 2013: (Start)
a(n) is the constant term of (2+x+y+1/x+1/y)^n.
G.f.: hypergeom([1/2, 1/2],[1],16*x^2/(1-2*x)^2)/(1-2*x). (End)
G.f.: 1 / AGM(1-6*x, 1+2*x), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - Paul D. Hanna, Aug 31 2014
D-finite with recurrence n^2*a(n) +2*(-3*n^2+3*n-1)*a(n-1) -4*(n-1)^2*a(n-2) +24*(n-1) *(n-2)*a(n-3)=0. - R. J. Mathar, Jun 14 2016
a(n) ~ 2^(n-1) * 3^(n+1) / (Pi*n). - Vaclav Kotesovec, Jul 20 2019
From Peter Bala, May 30 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n, 2*k) * binomial(2*k, k)^2 * 2^(n-2*k).
a(n) = 2^n * hypergeom([-n/2, (-n+1)/2, 1/2], [1, 1], 4). (End)
EXAMPLE
1 + 2*x + 8*x^2 + 32*x^3 + 148*x^4 + 712*x^5 + 3584*x^6 + 18496*x^7 + 97444*x^8 + ...
MAPLE
b:=n->coeff((x^2+1)^n, x, n); #A126869
bh:=n->add(binomial(n, k)*2^(n-k)*b(k)^2, k=0..n);
[seq(bh(n), n=0..30)];
# alternative program (faster for large n)
seq(simplify(2^n * hypergeom([-n/2, (-n+1)/2, 1/2], [1, 1], 4)), n = 0..30); # Peter Bala, May 30 2024
MATHEMATICA
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (Exp[x] BesselI[0, 2 x])^2, {x, 0, n}]] (* Michael Somos, Jun 27 2012 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( (exp(x + A) * besseli( 0, 2*x + A))^2, n))} /* Michael Somos, Jun 27 2012 */
(PARI) {a(n)=polcoeff( 1 / agm(1-6*x, 1+2*x +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 31 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 24 2011
STATUS
approved