A366115 - OEIS

%I #17 Aug 23 2024 10:49:42

%S 1,4,21,125,801,5390,37558,268656,1961355,14555266,109472688,

%T 832625469,6393072182,49488174700,385795571040,3026190911853,

%U 23867383581009,189156323865632,1505649098866535,12031665674394905,96486323017581420,776255276240140980

%N Expansion of (1/x) * Series_Reversion( x*(1+x+x^2)/(1+x)^5 ).

%H Seiichi Manyama, <a href="/A366115/b366115.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(4*n-k+4,n-2*k).

%F From _Peter Bala_, Aug 22 2024: (Start)

%F P-recursive: 6*n*(2*n+3)*(n^2-1)*(2021*n^3-8037*n^2+9946*n-3672)*a(n) = 4*n*(n-1)*(94987*n^5-282752*n^4+140624*n^3+146936*n^2-56373*n-18522)*a(n-1) - 6*(n-1)*(367822*n^6-1646645*n^5+2610582*n^4-1674935*n^3+259948*n^2+125940*n-35712)*a(n-2) + 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(2021*n^3-1974*n^2-65*n+258)*a(n-3) with a(0) = 1, a(1) = 4 and a(2) = 21.

%F G.f. A(x) satisfies 1 + x*A(x) = (1/x) * series_reversion( x/c(x*c(x)) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

%p seq(simplify(1/(n+1)*binomial(4*n+4, n)*hypergeom([n+1, -(1/2)*n, (1/2)*(1-n)], [3*n+5, -4*(n+1)], 4)), n = 0..20); # _Peter Bala_, Aug 22 2024

%o (PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(n+k, k)*binomial(4*n-k+4, n-2*k))/(n+1);

%Y Cf. A127632, A364374, A366114.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Sep 29 2023