A243157 Series reversion of x*(1 - x)/(1 - x - x^3).

1, 0, 0, -1, -1, -1, 2, 6, 11, 5, -21, -78, -124, -53, 335, 1096, 1727, 441, -5545, -17196, -25596, -2251, 97822, 284072, 399346, -44721, -1782873, -4876069, -6411063, 2201418, 33297536, 85893931, 104783903, -64745927, -632601621, -1541899544, -1727700472, 1642436289, 12171894474, 28062518974

Offset

1,7

Comments

Compare to A243156, where the g.f. G(x) satisfies:

x = G(x) * (1 - G(x)) / (1 - G(x) - G(x)^3) such that G(0) = 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

a(n) = (1/n)*(Sum_{k=0..floor((n-1)/3)} binomial(n,k)*binomial(n-2*k-2,n-1-3*k)*(-1)^k). - Tani Akinari, May 21 2018

D-finite with recurrence -n*(n+1)*a(n) +9*n*(n-1)*a(n-1) +6*(-6*n^2+22*n-17)*a(n-2) +27*(3*n-7)*(n-4)*a(n-3) +2*(-14*n^2+172*n-435)*a(n-4) -3*(53*n-192)*(n-5)*a(n-5) +341*(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Mar 24 2023

Example

G.f.: A(x) = x - x^4 - x^5 - x^6 + 2*x^7 + 6*x^8 + 11*x^9 + 5*x^10 - 21*x^11 - 78*x^12 - 124*x^13 - 53*x^14 + 335*x^15 +...
wherer A(x) = x * (1 - A(x) - A(x)^3) / (1 - A(x)).

Prog

(PARI) {a(n)=local(A=x); A=serreverse(x*(1 - x)/(1 - x - x^3 +x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, floor((n-1)/3), binomial(n, k)*binomial(n-2*k-2, n-1-3*k)*(-1)^k)/n} \\ Tani Akinari, May 21 2018

Crossrefs

Cf. A243156.

Sequence in context: A351716 A136699 A033710 * A274689 A123112 A092189

Adjacent sequences: A243154 A243155 A243156 * A243158 A243159 A243160

Keyword

sign

Author

Paul D. Hanna, May 31 2014

Status

approved