A182144 G.f. satisfies: A(x) = A(x^2)^2 + x*A(x)^2.

1, 1, 4, 9, 35, 104, 376, 1321, 4960, 18667, 72220, 282368, 1119791, 4481428, 18097960, 73612825, 301377323, 1240776848, 5133985196, 21337546123, 89037498752, 372879415520, 1566706843664, 6602445412864, 27900411735756, 118197671533743, 501897512293808

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

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

Let b(n) = Sum_{k=0..n} a(k)*a(n-k) form the self-convolution of this sequence, then

a(2*n+1) = b(2*n) for n>=0,

a(2*n) = b(2*n-1) + b(n) for n>0 with a(0)=1.

a(n) ~ c * d^n / n^(3/2), where d = 4.498712103893737093..., c = 0.7168247012663449... . - Vaclav Kotesovec, Aug 08 2014

Example

G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 35*x^4 + 104*x^5 + 376*x^6 +...
Related expansion:
A(x)^2 = 1 + 2*x + 9*x^2 + 26*x^3 + 104*x^4 + 350*x^5 + 1321*x^6 + 4856*x^7 + 18667*x^8 + 71870*x^9 + 282368*x^10 + 1118470*x^11 + 4481428*x^12 +...
From the coefficients in A(x)^2 we see that:
a(2) = 2 + 2 = 4; a(3) = 9;
a(4) = 9 + 26 = 35; a(5) = 104;
a(6) = 26 + 350 = 376; a(7) = 1321;
a(8) = 104 + 4856 = 4960; a(9) = 18667; ...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=subst(A, x, x^2+x*O(x^n))^2+x*A^2); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A182325, A181999.

Sequence in context: A006392 A179079 A084449 * A182723 A176607 A134815

Adjacent sequences: A182141 A182142 A182143 * A182145 A182146 A182147

Keyword

nonn

Author

Paul D. Hanna, Apr 24 2012

Status

approved