A265264 Terms satisfy: a(2*n) = a(n)*b(n) and a(2*n+1) = a(n+1)*b(n) for n>=0 with a(0)=1, where A(x)^2 = Sum_{n>=0} b(n)*x^n and g.f. A(x) = Sum_{n>=0} a(n)*x^n.

1, 1, 2, 4, 10, 20, 48, 120, 320, 640, 1520, 3648, 9216, 23040, 59520, 158720, 426240, 852480, 2037760, 4839680, 12038400, 28892160, 72142848, 182255616, 464781312, 1161953280, 2957967360, 7641415680, 19777781760, 52740751360, 138779688960, 372690616320, 990786355200, 1981572710400, 4793092669440, 11457362657280, 28548724162560

Offset

0,3

Comments

What is the limit a(n)^(1/n) = ? The limit is at least 2.535...

Links

Paul D. Hanna, Table of n, a(n) for n = 0..3000

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 20*x^5 + 48*x^6 + 120*x^7 + 320*x^8 + 640*x^9 + 1520*x^10 + 3648*x^11 + 9216*x^12 +...
where
A(x)^2 = 1 + 2*x + 5*x^2 + 12*x^3 + 32*x^4 + 76*x^5 + 192*x^6 + 496*x^7 + 1332*x^8 + 3184*x^9 + 7920*x^10 + 19776*x^11 + 50432*x^12 +...+ A265265(n)*x^n +...
Illustration of initial terms, both a(n) of A(x) and b(n) of A(x)^2:
a(0) = a(0)*b(0) = 1; b(0) = 1;
a(1) = a(1)*b(0) = 1; b(1) = 2 = 1*1 + 1*1;
a(2) = a(1)*b(1) = 2; b(2) = 5 = 1*2 + 1*1 + 2*1;
a(3) = a(2)*b(1) = 4; b(3) = 12 = 1*4 + 1*2 + 2*1 + 4*1;
a(4) = a(2)*b(2) = 10; b(4) = 32 = 1*10 + 1*4 + 2*2 + 4*1 + 10*1;
a(5) = a(3)*b(2) = 28; b(5) = 76;
a(6) = a(3)*b(3) = 48; b(6) = 192;
a(7) = a(4)*b(3) = 120; b(7) = 496;
a(8) = a(4)*b(4) = 320; b(8) = 1332; ...

Prog

(PARI) {a(n) = my(A=1+x); for(k=2, n, A = A + a((k+1)\2) * polcoeff(A^2, k\2) * x^k +x*O(x^n) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1, 1]); for(k=2, n, A = concat(A, A[(k+1)\2+1]*Vec(Ser(A)^2)[k\2+1]) ); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* Generates N terms rather quickly: */
N=300; A=[1, 1]; for(k=2, N, A = concat(A, A[(k+1)\2+1]*Vec(Ser(A)^2)[k\2+1]) ); A

Crossrefs

Cf. A265265 (A^2), A257889.

Sequence in context: A003407 A151523 A317708 * A026395 A247630 A015889

Adjacent sequences: A265261 A265262 A265263 * A265265 A265266 A265267

Keyword

nonn

Author

Paul D. Hanna, Dec 15 2015

Status

approved