A393660 - OEIS

%I #14 Mar 24 2026 09:44:39

%S 1,3,12,55,273,1428,7755,43335,247728,1442895,8537400,51198084,

%T 310618501,1903704213,11771483559,73361346135,460383173145,

%U 2907060507780,18457935937675,117776038227135,754846049949372,4857336644267848,31369745876798157,203259995090830620

%N Expansion of (1/x) * Series_Reversion( x * ( Sum_{k=0..5} (-x)^k )^3 ).

%H Vincenzo Librandi, <a href="/A393660/b393660.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1/x) * Series_Reversion( x * ((1-x^6) / (1+x))^3 ).

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/6)} binomial(3*n+k+2,k) * binomial(3*n+3,n-6*k).

%t CoefficientList[Normal@Series[InverseSeries@Series[x*((1-x^6)/(1+x))^3,{x,0,50}]/x,{x,0,20}],x] (* _Vincenzo Librandi_, Mar 24 2026 *)

%o (PARI) a(n) = sum(k=0, n\6, binomial(3*n+k+2, k)*binomial(3*n+3, n-6*k))/(n+1);

%o (Magma) N := 25; R<x> := PowerSeriesRing(Rationals(), N+5); f:= x*((1-x^6)/(1+x))^3; g:= Reverse(f) div x; Coeffs := [Coefficient(g,i):i in [0..N]]; Coeffs; // _Vincenzo Librandi_, Mar 24 2026

%Y Cf. A393658, A393659.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 24 2026