A208888 G.f. satisfies: A(x) = 1 + x*(A(x) + A(-x)) + x^2*A(x)*A(-x).

1, 2, 1, 2, -2, -4, -11, -22, -14, -28, 58, 116, 316, 632, 397, 794, -2198, -4396, -11954, -23908, -14684, -29368, 95170, 190340, 517492, 1034984, 623764, 1247528, -4462472, -8924944, -24270275, -48540550, -28820966, -57641932, 220608454, 441216908, 1200216340

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f.: A(x) = (1 - sqrt(1 - 4*x^2 + 16*x^4)) / (2*x^2*(1-2*x)).

Recurrence: (n+2)*a(n) = 2*(n+2)*a(n-1) + 4*(n-1)*a(n-2) - 8*(n-1)*a(n-3) - 16*(n-4)*a(n-4) + 32*(n-4)*a(n-5). - Vaclav Kotesovec, Aug 19 2013

|a(n)| ~ c * 3^(1/4)*2^(n+2)/(sqrt(Pi)*n^(3/2)), where c=(sqrt(3)+1)/2 if n=6k+0 or n=6k+1, c=(sqrt(3)-1)/2 if n=6k+2 or n=6k+3 and c=1 if n=6k+4 or n=6k+5. - Vaclav Kotesovec, Aug 19 2013

Example

G.f.: A(x) = 1 + 2*x + x^2 + 2*x^3 - 2*x^4 - 4*x^5 - 11*x^6 - 22*x^7 +...
Related series:
A(x)+A(-x) = 2 + 2*x^2 - 4*x^4 - 22*x^6 - 28*x^8 + 116*x^10 + 632*x^12 +...
A(x)*A(-x) = 1 - 2*x^2 - 11*x^4 - 14*x^6 + 58*x^8 + 316*x^10 + 397*x^12 +...

Maple

A208888_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1] - add((-1)^j*a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list); subsop(1=NULL, %); end: A208888_list(37); # Peter Luschny, Feb 29 2016

Mathematica

CoefficientList[Series[(1-Sqrt[1-4*x^2+16*x^4])/(2*x^2*(1-2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 19 2013 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*(A+subst(A, x, -x+x*O(x^n)))+x^2*A*subst(A, x, -x+x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A208887.

Sequence in context: A193691 A089408 A350287 * A258783 A079318 A050315

Adjacent sequences: A208885 A208886 A208887 * A208889 A208890 A208891

Keyword

sign

Author

Paul D. Hanna, Mar 09 2012

Status

approved