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A144904 Coefficient of x^n in expansion of x/((1-x-x^3)*(1-x)^(n-1)), also diagonal of A144903.

%I A144904 #23 Apr 08 2024 14:40:52
%S A144904 0,1,2,6,21,76,280,1045,3937,14938,56993,218414,840090,3241153,
%T A144904 12537263,48604755,188799962,734631798,2862843281,11171582151,
%U A144904 43647688211,170720728344,668414462009,2619400928928,10273572796046,40325085206853,158393604268277
%N A144904 Coefficient of x^n in expansion of x/((1-x-x^3)*(1-x)^(n-1)), also diagonal of A144903.
%H A144904 Alois P. Heinz, <a href="/A144904/b144904.txt">Table of n, a(n) for n = 0..500</a>
%F A144904 a(n) = [x^n] x/((1-x-x^3)*(1-x)^(n-1)).
%F A144904 From _G. C. Greubel_, Jul 27 2022: (Start)
%F A144904 a(n) = Sum_{j=0..floor((n-1)/3)} binomial(2*n-2*j-2, n+j-1).
%F A144904 a(n) = A099567(2*n, n). (End)
%F A144904 a(n) = binomial(2*(n-1), n-1)*hypergeom([1, (1-n)/3, (2-n)/3, 1-n/3], [1-n, 3/2-n, n], -27/4) for n > 0. - _Stefano Spezia_, Apr 06 2024
%F A144904 a(n) ~ 4^n/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 08 2024
%p A144904 A:= proc(n,k) coeftayl (x/ (1-x-x^3)/ (1-x)^(k-1), x=0, n) end:
%p A144904 a:= n-> A(n,n):
%p A144904 seq(a(n), n=0..30);
%p A144904 # second Maple program:
%p A144904 a:= proc(n) option remember; `if`(n<3, n,
%p A144904       ((27*n^3-150*n^2+195*n-12)*a(n-1)
%p A144904       -(66*n^3-382*n^2+492*n+124)*a(n-2)
%p A144904       +(27*n^3-156*n^2+201*n+48)*a(n-3)
%p A144904       -2*(2*n-7)*(3*n^2-7*n-2)*a(n-4))/((n-1)*(3*n^2-13*n+8)))
%p A144904     end:
%p A144904 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 06 2013
%t A144904 Table[Sum[Binomial[2*n-2*j-2, n+j-1], {j,0,Floor[(n-1)/3]}], {n,0,40}] (* _G. C. Greubel_, Jul 27 2022 *)
%o A144904 (Magma)
%o A144904 A144904:= func< n | n eq 0 select 0 else  (&+[Binomial(2*n-2*j-2, n+j-1): j in [0..Floor((n-1)/3)]]) >;
%o A144904 [A144904(n): n in [0..40]]; // _G. C. Greubel_, Jul 27 2022
%o A144904 (SageMath)
%o A144904 def A144904(n): return sum(binomial(2*n-2*j-2, n+j-1) for j in (0..((n-1)//3)))
%o A144904 [A144904(n) for n in (0..40)] # _G. C. Greubel_, Jul 27 2022
%Y A144904 Cf. A099567, A144903.
%K A144904 nonn
%O A144904 0,3
%A A144904 _Alois P. Heinz_, Sep 24 2008