cp's OEIS Frontend

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

%I A243157 #14 Mar 24 2023 14:54:46
%S A243157 1,0,0,-1,-1,-1,2,6,11,5,-21,-78,-124,-53,335,1096,1727,441,-5545,
%T A243157 -17196,-25596,-2251,97822,284072,399346,-44721,-1782873,-4876069,
%U A243157 -6411063,2201418,33297536,85893931,104783903,-64745927,-632601621,-1541899544,-1727700472,1642436289,12171894474,28062518974
%N A243157 Series reversion of x*(1 - x)/(1 - x - x^3).
%C A243157 Compare to A243156, where the g.f. G(x) satisfies:
%C A243157 x = G(x) * (1 - G(x)) / (1 - G(x) - G(x)^3) such that G(0) = 1.
%H A243157 Paul D. Hanna, <a href="/A243157/b243157.txt">Table of n, a(n) for n = 1..300</a>
%F A243157 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
%F A243157 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
%e A243157 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 +...
%e A243157 wherer A(x) = x * (1 - A(x) - A(x)^3) / (1 - A(x)).
%o A243157 (PARI) {a(n)=local(A=x);A=serreverse(x*(1 - x)/(1 - x - x^3 +x*O(x^n)));polcoeff(A,n)}
%o A243157 for(n=0,40,print1(a(n),", "))
%o A243157 (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
%Y A243157 Cf. A243156.
%K A243157 sign
%O A243157 1,7
%A A243157 _Paul D. Hanna_, May 31 2014