OFFSET
0,2
COMMENTS
a(n) = A323557(n*(n+2)) for n >= 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..179
FORMULA
a(n) = [x^(n*(n+2))] Sum_{k>=0} x^k * (1 + x^k)^k / (1 + x^(k+1))^(k+1).
a(n) = [x^(n*(n+2))] Sum_{k>=0} (-x)^k * (1 - x^k)^k / (1 - x^(k+1))^(k+1).
EXAMPLE
Given the g.f. of A323557, G(x) = Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1), i.e.,
G(x) = 1/(1 + x) + x*(1 + x)/(1 + x^2)^2 + x^2*(1 + x^2)^2/(1 + x^3)^3 + x^3*(1 + x^3)^3/(1 + x^4)^4 + x^4*(1 + x^4)^4/(1 + x^5)^5 + x^5*(1 + x^5)^5/(1 + x^6)^6 + x^6*(1 + x^6)^6/(1 + x^7)^7 + x^7*(1 + x^7)^7/(1 + x^8)^8 + ...
and writing G(x) as a power series in x starting as
G(x) = 1 + 3*x^2 - 2*x^3 + 2*x^4 + 9*x^6 - 14*x^7 + 8*x^8 + 12*x^10 - 12*x^11 + 15*x^12 - 52*x^13 + 76*x^14 - 36*x^15 + 2*x^16 + 50*x^18 - 104*x^19 + 79*x^20 - 140*x^21 + 324*x^22 - 276*x^23 + 128*x^24 - 144*x^25 + 118*x^26 - 28*x^27 + 72*x^28 - 336*x^29 + 657*x^30 - 802*x^31 + 1184*x^32 - 1568*x^33 + 1086*x^34 - 288*x^35 + 302*x^36 - 1032*x^37 + 1212*x^38 - 480*x^39 + 142*x^40 + ...
then the coefficients of x^(n*(n+2)) in G(x), for n >= 0, form this sequence.
PROG
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 04 2019
STATUS
approved