A372233 - OEIS

%I #47 May 04 2024 04:46:09

%S 1,2,12,77,520,3612,25557,183192,1325808,9666635,70897112,522472392,

%T 3865669717,28697325048,213649228560,1594540806612,11926354293792,

%U 89372808145692,670865679851667,5043360211505000,37965778448487120,286151354441445570,2159143860124095120

%N Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x-x^2) )^n.

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(3*n-k-1,n-2*k).

%F The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x) * (1-x-x^2) ).

%F D-finite with recurrence +575*n*(n-1)*(n-2)*a(n) +40*(n-1)*(n-2)*(125*n-178)*a(n-1) -16*(n-2)*(3272*n^2-5536*n+75)*a(n-2) +8*(-22112*n^3+169392*n^2-450082*n+415827)*a(n-3) +1344*(96*n^3-1328*n^2+5794*n-8139)*a(n-4) +3072*(4*n-15)*(2*n-9)*(4*n-17)*a(n-5)=0. - _R. J. Mathar_, May 02 2024

%F a(n) ~ sqrt((1/8 + cos(arccos(sqrt(37)/8)/3)/sqrt(37))/(Pi*n)) / (-2/3 + sqrt(35/18)*cos(arccos(-4537/(560*sqrt(70)))/3))^n. - _Vaclav Kotesovec_, May 04 2024

%p A372233 := proc(n)

%p     add(binomial(n+k-1,k) * binomial(3*n-k-1,n-2*k),k=0..floor(n/2));

%p end proc:

%p seq(A372233(n),n=0..50) ; # _R. J. Mathar_, May 02 2024

%t Table[SeriesCoefficient[1/((1-x)*(1-x-x^2))^n, {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, May 04 2024 *)

%o (PARI) a(n, s=2, t=1, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t+u+1)*n-(s-1)*k-1, n-s*k));

%Y Cf. A213684, A236339, A372458, A372460.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 02 2024