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A370280 Coefficient of x^n in the expansion of 1/( (1-x)^2 - x )^n.

%I A370280 #14 Feb 07 2025 09:26:40
%S A370280 1,3,25,234,2305,23373,241486,2527920,26720529,284555700,3048323135,
%T A370280 32812937820,354619072990,3845377105794,41817926091120,
%U A370280 455893204069944,4980851709418353,54521955043418925,597823622561048020,6564929893462467450,72189820135528858455
%N A370280 Coefficient of x^n in the expansion of 1/( (1-x)^2 - x )^n.
%H A370280 G. C. Greubel, <a href="/A370280/b370280.txt">Table of n, a(n) for n = 0..940</a>
%F A370280 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(3*n+k-1,n-k).
%F A370280 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * ((1-x)^2 - x) ).
%F A370280 a(n) ~ sqrt((4 + sqrt(6))/(24*Pi*n)) * ((27 + 12*sqrt(6))/5)^n. - _Vaclav Kotesovec_, Feb 07 2025
%t A370280 A370280[n_]:= Coefficient[Series[1/(1-3*x+x^2)^n, {x,0,100}], x, n];
%t A370280 Table[A370280[n], {n,0,40}] (* _G. C. Greubel_, Feb 07 2025 *)
%o A370280 (PARI) a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(3*n+k-1, n-k));
%o A370280 (Magma)
%o A370280 R<x>:=PowerSeriesRing(Rationals(), 100);
%o A370280 A370280:= func< n | Coefficient(R!( 1/(1-3*x+x^2)^n ), n) >;
%o A370280 [A370280(n): n in [0..30]]; // _G. C. Greubel_, Feb 07 2025
%o A370280 (SageMath)
%o A370280 def A370280(n): return sum(binomial(n+j-1,j)*binomial(3*n+j-1,n-j) for j in range(n+1))
%o A370280 print([A370280(n) for n in range(31)]) # _G. C. Greubel_, Feb 07 2025
%Y A370280 Cf. A069723, A249924, A370282.
%K A370280 nonn
%O A370280 0,2
%A A370280 _Seiichi Manyama_, Feb 13 2024