A202146 G.f.: 1/(1-x) + Sum_{n>=1} x^n/(1-x) * Product_{k=1..n} (1 - x^k) / (1 - x^(2*k+1)).

1, 2, 2, 2, 2, 2, 1, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 1, 2, 2, 0, 2, 4, 0, 2, 2, 0, 4, 2, 0, 2, 2, 2, 2, 2, -1, 2, 4, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 2, 0, 0, 2, 4, 0, 2, 2, -2, 4, 2, 1, 2, 2, 2, 0, 2, 0, 2, 4, 2, 2, 0, 0, 4, 2, 0, 2, 2, 2, 2, 2, 0, 2, 4, 0, 0

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2106

Formula

a(k) == 1 (mod 2) iff k = 3*n*(n+1) for n>=0 (conjecture).

Example

G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + x^6 + 2*x^7 +...
where A(x) = 1/(1-x) + x*(1-x)/((1-x)*(1-x^3)) + x^2*(1-x)*(1-x^2)/((1-x)*(1-x^3)*(1-x^5)) + x^3*(1-x)*(1-x^2)*(1-x^3)/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)) +...
which is a q-series analog of the sum:
Pi/2 = 0!/1 + 1!/(1*3) + 2!/(1*3*5) + 3!/(1*3*5*7) + 4!/(1*3*5*7*9) + 5!/(1*3*5*7*9*11) +...
Odd terms (A202150), located at positions 3*n*(n+1) for n>=0, begin:
[1,1,1,-1,1,-1,1,3,1,-1,1,-1,1,1,1,-1,-1,1,1,1,1,-1,3,-1,1,1,1,...].

Prog

(PARI) {a(n)=polcoeff((1+sum(m=1, n, x^m*prod(k=1, m, (1-x^k)/(1-x^(2*k+1) +x*O(x^n)))))/(1-x+x*O(x^n)), n)}

Crossrefs

Cf. A202145 (first differences), A202150 (odd terms).

Sequence in context: A027386 A102300 A359511 * A087010 A098220 A297032

Adjacent sequences: A202143 A202144 A202145 * A202147 A202148 A202149

Keyword

sign

Author

Paul D. Hanna, Dec 12 2011

Status

approved