A121648 a(n) is the square of the coefficient of x^n in 1/(1 - x*A(x^2)), where g.f. A(x) = Sum_{n>=0} a(n)*x^n.

1, 1, 1, 4, 9, 25, 64, 256, 729, 2601, 7921, 28900, 90000, 318096, 1016064, 3888784, 12694969, 46881409, 155825289, 592435600, 1987643889, 7408217041, 25153960000, 95823678916, 327811212304, 1227814692624, 4234507915264

Offset

0,4

Links

Table of n, a(n) for n=0..26.

Formula

a(n) = A121649(n)^2.

G.f.: A(x^2) = (1 - 1/B(x) )/x, where B(x) = Sum_{n>=0} a(n)^(1/2)*x^n is the g.f. of A121649.

Example

A(x) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 25*x^5 + 64*x^6 + 256*x^7 +...
Take the square root of each term, a(n)^(1/2), and
let B(x) be the g.f. of the resulting sequence:
B(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 16*x^7 +...
Then 1/B(x) = 1 - x*A(x^2):
1/B(x) = 1 - x - x^3 - x^5 - 4*x^7 - 9*x^9 - 25*x^11 - 64*x^13 -...

Prog

(PARI) {a(n)=if(n==0, 1, sum(k=0, n\2, polcoeff(x^(n-2*k)*(sum(j=0, k, a(j)*x^(2*j))+x*O(x^n))^(n-2*k), n))^2)}
(PARI) {a(n)=local(A, m); if(n<0, 0, m=1; A=1+x+O(x^2); while(m<=n, m*=2; A=1/(1-x*sum(k=0, m-1, polcoeff(A, k)^2*x^(2*k), O(x^(2*m))))); polcoeff(A, n)^2)} /* Michael Somos, Aug 18 2006 */

Crossrefs

Cf. A121649; bisection of A121649: A121650, A121651.

Sequence in context: A244558 A175627 A007598 * A133022 A184326 A217590

Adjacent sequences: A121645 A121646 A121647 * A121649 A121650 A121651

Keyword

nonn

Author

Paul D. Hanna, Aug 14 2006

Status

approved