A357406 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n * x^(2*n+2) * (1 - x^n)^(n+1).

1, 0, -1, 0, 3, -8, 9, 0, -10, 0, 24, -24, 0, 0, 15, 0, 9, -80, 90, 0, -43, 0, 57, -80, 13, 0, 175, -200, 15, -120, 313, 0, -346, 0, 450, -168, 19, -744, 830, 0, 21, -224, -287, 0, 405, 0, 1014, -1968, 25, 0, 2813, -784, -2448, -360, 1575, 0, 2765, -3520, 450, -440, 31

Offset

0,5

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2050

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) A(x) = Sum_{n=-oo..+oo} n * x^(2*n+2) * (1 - x^n)^(n+1),

(2) A(x) = -Sum_{n=-oo..+oo, n<>0} n * (-1)^n * x^((n-1)*(n-2)) / (1 - x^n)^(n-1).

Example

G.f.: A(x) = 1 - x^2 + 3*x^4 - 8*x^5 + 9*x^6 - 10*x^8 + 24*x^10 - 24*x^11 + 15*x^14 + 9*x^16 - 80*x^17 + 90*x^18 - 43*x^20 + 57*x^22 - 80*x^23 + 13*x^24 + ...
Related series.
x/A(x) = x + x^3 - 2*x^5 + 8*x^6 - 14*x^7 + 16*x^8 - 7*x^9 - 24*x^10 + 103*x^11 - 232*x^12 + 334*x^13 - 256*x^14 - 211*x^15 + 1400*x^16 + ... + A357401(n)*x^n + ...

Prog

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

Crossrefs

Cf. A357401, A356774.

Sequence in context: A161432 A229380 A229372 * A021261 A245263 A016672

Adjacent sequences: A357403 A357404 A357405 * A357407 A357408 A357409

Keyword

sign

Author

Paul D. Hanna, Sep 27 2022

Status

approved