A340775 G.f.: Sum_{n>=0} x^n/(1 - x^2*(1+x)^n).

1, 1, 2, 2, 4, 5, 10, 16, 32, 61, 127, 270, 600, 1378, 3274, 8021, 20245, 52535, 140014, 382745, 1072025, 3073443, 9010934, 26994231, 82563483, 257634875, 819648796, 2656956702, 8770406923, 29464217659, 100689885448, 349849796512

Offset

0,3

Comments

The g.f. of this sequence is motivated by the following identity:

Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1 - q*r^n)) ;

here, p = x, q = x^2, and r = (1+x).

Links

Table of n, a(n) for n=0..31.

Formula

G.f.: Sum_{n>=0} x^n / (1 - x^2*(1+x)^n).

G.f.: Sum_{n>=0} x^(2*n) / (1 - x*(1+x)^n).

G.f.: Sum_{n>=0} x^(3*n) * (1+x)^(n^2) * (1 - x^3*(1+x)^(2*n)) / ((1 - x*(1+x)^n)*(1 - x^2*(1+x)^n)).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 10*x^6 + 16*x^7 + 32*x^8 + 61*x^9 + 127*x^10 + 270*x^11 + 600*x^12 + 1378*x^13 + 3274*x^14 + 8021*x^15 + ...
where
A(x) = 1/(1 - x^2) + x/(1 - x^2*(1+x)) + x^2/(1 - x^2*(1+x)^2) + x^3/(1 - x^2*(1+x)^3) + x^4/(1 - x^2*(1+x)^4) + x^5/(1 - x^2*(1+x)^5) + ...
also
A(x) = 1/(1-x) + x^2/(1 - x*(1+x)) + x^4/(1 - x*(1+x)^2) + x^6/(1 - x*(1+x)^3) + x^8/(1 - x*(1+x)^4) + x^10/(1 - x*(1+x)^5) + ...

Prog

(PARI) {a(n) = my(A = sum(m=0, n, x^m /(1 - x^2*(1+x)^m +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n, x^(2*m) /(1 - x*(1+x)^m +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A340776.

Sequence in context: A054538 A238020 A095020 * A290436 A338048 A127825

Adjacent sequences: A340772 A340773 A340774 * A340776 A340777 A340778

Keyword

nonn

Author

Paul D. Hanna, Jan 20 2021

Status

approved