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 A369299 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x^3)^3 ). 5

%I #20 Feb 15 2024 04:11:37

%S 1,1,2,8,29,105,417,1719,7181,30603,132736,582790,2585352,11575613,

%T 52237278,237328704,1084701387,4983867447,23007263941,106658256768,

%U 496336303014,2317687534865,10856677523580,51001805706435,240225121539000,1134240896062656,5367428039668751

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

%H Seiichi Manyama, <a href="/A369299/b369299.txt">Table of n, a(n) for n = 0..1000</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(3*n+k+2,k) * binomial(2*n-3*k,n-3*k).

%F a(n) = (1/(n+1)) * [x^n] 1/( (1-x) * (1-x^3)^3 )^(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)^3)/x)

%o (PARI) a(n, s=3, t=3, u=1) = 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. A369300, A369301.

%Y Cf. A063030, A369296.

%Y Cf. A369268.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 18 2024

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Last modified August 8 08:46 EDT 2024. Contains 375018 sequences. (Running on oeis4.)