A257889 Terms satisfy: a(2*n) = a(n)*b(n) and a(2*n+1) = a(n)*b(n+1) 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, 5, 10, 28, 70, 170, 340, 960, 2688, 7308, 18270, 48440, 117640, 285600, 571200, 1543600, 4358400, 12038400, 33707520, 92875776, 252506016, 677656224, 1694140560, 4596220440, 12186147680, 32083749600, 77917677600, 203473437920, 493981756800, 1217092396800, 2434184793600, 6357547392000, 17180514976000, 46002675920000, 129889908480000, 349123704576000

Offset

0,3

Comments

5 divides a(3*2^n) for n>=0 (conjecture).

5 does not divide a(3*2^n - 1) for n>=0 (conjecture).

What is the limit a(n)^(1/n) = ?

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2500

Vaclav Kotesovec, Plot of a(n+1)/a(n)

Example

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 28*x^5 + 70*x^6 + 170*x^7 + 340*x^8 + 960*x^9 + 2688*x^10 + 7308*x^11 + 18270*x^12 +...
where
A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 34*x^4 + 96*x^5 + 261*x^6 + 692*x^7 + 1680*x^8 + 4540*x^9 + 12540*x^10 + 34552*x^11 + 92728*x^12 +...+ A265226(n)*x^n +...
Illustration of initial terms, both a(n) of A(x) and b(n) of A(x)^2:
a(0) = 1; b(0) = 1;
a(1) = 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(1)*b(2) = 5; b(3) = 14 = 1*5 + 1*2 + 2*1 + 5*1;
a(4) = a(2)*b(2) = 10; b(4) = 34 = 1*10 + 1*5 + 2*2 + 5*1 + 10*1;
a(5) = a(2)*b(3) = 28; b(5) = 96;
a(6) = a(3)*b(3) = 70; b(6) = 261;
a(7) = a(3)*b(4) = 170; b(7) = 692;
a(8) = a(4)*b(4) = 340; b(8) = 1680; ...

Prog

(PARI) {a(n) = my(A=1+x); for(k=2, n, A = A + a(k\2) * polcoeff(A^2, (k+1)\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\2+1]*Vec(Ser(A)^2)[(k+1)\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\2+1]*Vec(Ser(A)^2)[(k+1)\2+1]) ); A

Crossrefs

Cf. A265226 (A^2), A265264, A264927.

Sequence in context: A074801 A324838 A370316 * A363388 A287963 A006401

Adjacent sequences: A257886 A257887 A257888 * A257890 A257891 A257892

Keyword

nonn

Author

Paul D. Hanna, Dec 13 2015

Status

approved