A368079 - OEIS

%I #42 Feb 16 2024 09:51:55

%S 1,3,18,127,996,8322,72644,654615,6043455,56866028,543368586,

%T 5258196762,51426990112,507537393600,5048033356128,50549237164615,

%U 509197913456922,5156339940802941,52460340305220466,535976129228082972,5496745175387480976

%N Expansion of (1/x) * Series_Reversion( x * (1-x)^3 * (1-x^2)^3 ).

%H Seiichi Manyama, <a href="/A368079/b368079.txt">Table of n, a(n) for n = 0..963</a>

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+k+2,k) * binomial(4*n-2*k+2,n-2*k).

%F a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(3*n+k+2,k) * binomial(7*n-k+5,n-k).

%F a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^3 * (1-x)^6 )^(n+1). - _Seiichi Manyama_, Feb 16 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3*(1-x^2)^3)/x)

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

%Y Cf. A365752, A365856.

%Y Cf. A365878, A365879.

%Y Cf. A130565, A369301.

%Y Cf. A369264, A370271.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 18 2024