A323677 Coefficients of x^(n*(n+2)) in Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1).

1, -2, 8, -36, 128, -288, 1166, -16048, 77328, -108780, 220440, -5900816, 44395366, -339891804, 898603106, -5623621248, 2160154604, -22569496436, 930080367404, -11435095847280, 54170773716302, -95118392809132, 264120000910222, -449060398964736, 10522002570379158, -174697886314749868, 1461399560338555304, -10156216740712605688, 35485949488006257078, -90157116358607006060, 722914685477655291098

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

(PARI) {A323557(n) = my(A=sum(m=0, n, x^m * (1 + x^m +x*O(x^n))^m/(1 + x^(m+1) +x*O(x^n))^(m+1) )); polcoeff(A, n)}
a(n) = A323557(n*(n+2))
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A323557, A323678, A323679.

Sequence in context: A355102 A355104 A248861 * A295501 A283847 A123290

Adjacent sequences: A323674 A323675 A323676 * A323678 A323679 A323680

Keyword

sign

Author

Paul D. Hanna, Feb 04 2019

Status

approved