A350435 G.f. A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1+x, F(2) = 1 + x*(1+x), and F(n+1) = 1 + (F(n-1) - 1)*(F(n) - 1)*F(n) for n > 1.

1, 2, 3, 6, 13, 25, 47, 97, 202, 405, 804, 1620, 3317, 6892, 14473, 30380, 63169, 129986, 265696, 541597, 1104411, 2258693, 4640651, 9583818, 19888842, 41440324, 86582460, 181122452, 378791172, 791010101, 1648083517, 3425009503, 7100406046, 14690106611, 30348419552

Offset

1,2

Links

Table of n, a(n) for n=1..35.

Example

G.f.: A(x) = x + 2*x^2 + 3*x^3 + 6*x^4 + 13*x^5 + 25*x^6 + 47*x^7 + 97*x^8 + 202*x^9 + 405*x^10 + 804*x^11 + 1620*x^12 + ...
GENERATING METHOD.
Define F(n), a polynomial in x of order Pell(n), by the following recurrence involving Fibonacci numbers:
F(0) = x,
F(1) = (1 + x),
F(2) = (1 + x * (1+x)),
F(3) = (1 + x^2 * (1+x) * F(2)),
F(4) = (1 + x^3 * (1+x)^2 * F(2) * F(3)),
F(5) = (1 + x^5 * (1+x)^3 * F(2)^2 * F(3) * F(4)),
F(6) = (1 + x^8 * (1+x)^5 * F(2)^3 * F(3)^2 * F(4) * F(5)),
F(7) = (1 + x^13 * (1+x)^8 * F(2)^5 * F(3)^3 * F(4)^2 * F(5) * F(6)),
...
F(n+1) = 1 + (F(n-1) - 1) * (F(n) - 1) * F(n)
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
that is,
A(x) = x * (1+x) * (1 + x*(1+x)) * (1 + x^2*(1+x)*(1 + x*(1+x))) * (1 + x^3*(1+x)^2*(1 + x*(1+x))*(1 + x^2*(1+x)*(1 + x*(1+x)))) * ...

Prog

(PARI) {F(n) = my(G=x); if(n==0, G=x, if(n==1, G=1+x, if(n==2, G=1+x*(1+x), G = 1 + (F(n-2) - 1)*(F(n-1) - 1)*F(n-1)))); G}
{a(n) = my(A = prod(k=0, round(log(n)/log(3/2))+1, F(k) +x*O(x^n))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A350432.

Sequence in context: A239326 A075530 A032061 * A155996 A240104 A018274

Adjacent sequences: A350432 A350433 A350434 * A350436 A350437 A350438

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2021

Status

approved