A323696 a(n) = [x^(n*(n+2))] Sum_{m>=0} x^m * (1 - x^(m+1))^m / (1 + x^(m+1))^(m+1).

1, -3, 15, -93, 323, -1753, 8991, -58747, 268541, -1462761, 8097455, -76494807, 251595971, -1409934009, 9674895101, -46242141689, 252055236611, -1426069096367, 8079317057871, -46569083263127, 260949012214457, -1482376214228889, 8443414161166175, -90551791833348977, 274738209468977925, -1569245074591691433, 8970254867527278283, -51444993272705476951, 293733710358893793731, -1737504244377865756267, 9642641465118083682431, -55353191176361412679671

Offset

0,2

Comments

a(n) = A323695(n*(n+2)) for n >= 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

a(n) = [x^(n*(n+2))] Sum_{n>=0} x^n * (1 - x^(n+1))^n / (1 + x^(n+1))^(n+1).

a(n) = [x^(n*(n+2))] Sum_{n>=0} (-x)^n * (1 + x^(n+1))^n / (1 - x^(n+1))^(n+1).

Example

Given the g.f. of A323675, G(x) = Sum_{n>=0} x^n * (1 - x^(n+1))^n / (1 + x^(n+1))^(n+1), i.e.,
G(x) =  1/(1+x) + x*(1-x^2)/(1+x^2)^2 + x^2*(1-x^3)^2/(1+x^3)^3 + x^3*(1-x^4)^3/(1+x^4)^4 + x^4*(1-x^5)^4/(1+x^5)^5 + x^5*(1-x^6)^5/(1+x^6)^6 + x^6*(1-x^7)^6/(1+x^7)^7 + x^7*(1-x^8)^7/(1+x^8)^8 + x^8*(1-x^9)^8/(1+x^9)^9 + ...
and writing G(x) as a power series in x starting as
G(x) = 1 + 2*x^2 - 3*x^3 + 2*x^4 + 2*x^6 - 14*x^7 + 15*x^8 + 2*x^10 - 22*x^11 + 2*x^12 + 84*x^14 - 93*x^15 + 2*x^16 + 2*x^18 - 38*x^19 + 172*x^20 + 2*x^22 - 508*x^23 + 323*x^24 + 292*x^26 - 54*x^27 + 2*x^28 + 2*x^30 - 1212*x^31 + 444*x^32 + 2580*x^34 - 1753*x^35 + ...
then the odd coefficients of x^n in G(x), occurring at n = k*(k+2) for k>=0, form this sequence.

Prog

(PARI) {A323695(n) = my(SUM = sum(m=0, n, x^m*(1 - x^(m+1) +x*O(x^n))^m / (1 + x^(m+1) +x*O(x^n))^(m+1) )); polcoeff(SUM, n)}
{a(n) = A323695(n*(n+2))}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A323695.

Cf. A324303, A323676.

Sequence in context: A366053 A369114 A020108 * A231657 A361881 A303558

Adjacent sequences: A323693 A323694 A323695 * A323697 A323698 A323699

Keyword

sign

Author

Paul D. Hanna, Feb 23 2019

Status

approved