A369266 - OEIS

%I #17 Feb 15 2024 04:18:41

%S 1,1,2,7,24,84,313,1209,4769,19166,78253,323570,1352122,5701467,

%T 24229122,103663575,446163435,1930390329,8391341664,36630504952,

%U 160509484616,705750073063,3112865367660,13769327908980,61066953746400,271488240652950,1209671359828154

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

%H P. Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</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/3)} binomial(2*n+2,k) * binomial(2*n-3*k,n-3*k).

%F a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^3)^2 )^(n+1). - _Seiichi Manyama_, Feb 14 2024

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

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

%Y Cf. A071969, A369268.

%Y Cf. A369267, A370248.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 18 2024