A322618 G.f.: Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x + x^(n+1))^(n+1).

1, -1, 4, -11, 24, -47, 95, -204, 440, -915, 1829, -3556, 6819, -13043, 25039, -48306, 93509, -181219, 351089, -679814, 1316526, -2552828, 4961602, -9672267, 18916895, -37114623, 73027149, -144047576, 284741852, -563872928, 1118354547, -2221007241, 4415827590, -8788137259, 17504252965, -34889806960, 69584723922, -138850448656, 277179930085, -553510258815, 1105635941562

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

G.f.: Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x + x^(n+1))^(n+1).

G.f.: Sum_{n>=0} (-x)^n * (1 - x^n)^n / (1 + x - x^(n+1))^(n+1).

Example

G.f.: A(x) = 1 - x + 4*x^2 - 11*x^3 + 24*x^4 - 47*x^5 + 95*x^6 - 204*x^7 + 440*x^8 - 915*x^9 + 1829*x^10 - 3556*x^11 + 6819*x^12 - 13043*x^13 + 25039*x^14 + ...
such that
A(x) = 1/(1 + 2*x) + x*(1+x)/(1+x + x^2)^2 + x^2*(1+x^2)^2/(1+x + x^3)^3 + x^3*(1+x^3)^3/(1+x + x^4)^4 + x^4*(1+x^4)^4/(1+x + x^5)^5 + x^5*(1+x^5)^5/(1+x + x^6)^6 + ...
also,
A(x) = 1 - x*(1-x)/(1+x - x^2)^2 + x^2*(1-x^2)^2/(1+x - x^3)^3 - x^3*(1-x^3)^3/(1+x - x^4)^4 + x^4*(1-x^4)^4/(1+x - x^5)^5 - x^5*(1-x^5)^5/(1+x - x^6)^6 + ...

Prog

(PARI) {a(n) = my(A = sum(m=0, n, x^m*(1 + x^m)^m/(1 + x + x^(m+1) +x*O(x^n) )^(m+1) ) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n, (-x)^m*(1 - x^m)^m/(1 + x - x^(m+1) +x*O(x^n) )^(m+1) ) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

Crossrefs

Cf. A322619.

Sequence in context: A192748 A143075 A290707 * A260057 A260150 A258472

Adjacent sequences: A322615 A322616 A322617 * A322619 A322620 A322621

Keyword

sign

Author

Paul D. Hanna, Jan 26 2019

Status

approved