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A127905 Construct triangle in which n-th row is obtained by expanding (1+x+x^3)^n and take the next-to-central column.

%I A127905 #17 Sep 08 2022 08:45:29
%S A127905 0,1,2,3,8,25,66,168,456,1269,3490,9581,26544,73944,206220,576045,
%T A127905 1613264,4527661,12725946,35818135,100950440,284869263,804726934,
%U A127905 2275500998,6440230392,18242735800,51714552656
%N A127905 Construct triangle in which n-th row is obtained by expanding (1+x+x^3)^n and take the next-to-central column.
%H A127905 G. C. Greubel, <a href="/A127905/b127905.txt">Table of n, a(n) for n = 0..1000</a>
%F A127905 a(n) = n*A071879(n-1).
%F A127905 a(n) = n*Sum_{k=0..floor((n-1)/3)} C(n-1,3*k)*C(3*k,k)/(2*k+1).
%F A127905 a(n) = Sum_{k=0..floor((n-1)/3)} (3*k+1)*C(n,3*k+1)*C(3*k,k)/(2k+1).
%F A127905 a(n) = Sum_{k=0..n-1} Sum_{j=0..floor(k/3)} C(k,3*j)*C(3*j+1,j).
%F A127905 Conjecture: 2*(2*n+1)*(n-1)^2*a(n) -2*n*(6*n^2-12*n+5)*a(n-1) +6*n*(n-1)*(2*n-3)*a(n-2) -31*n*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Feb 23 2015
%F A127905 a(n) ~ (1 + 3/2^(2/3))^(n + 1/2) / sqrt(12*Pi*n). - _Vaclav Kotesovec_, May 01 2018
%p A127905 A127905 := proc(n)
%p A127905     n*add(binomial(n-1,3*k)*binomial(3*k,k)/(2*k+1),k=0..floor((n-1)/3)) ;
%p A127905 end proc: # _R. J. Mathar_, Feb 23 2015
%t A127905 Table[n*Sum[Binomial[n-1,3*k]*Binomial[3*k,k]/(2*k+1), {k, 0, Floor[(n -1)/3]}], {n, 0, 50}] (* _G. C. Greubel_, Apr 30 2018 *)
%o A127905 (PARI) a(n)=if(n<0, 0, polcoeff((1+x+x^3)^n, n-1));
%o A127905 (PARI) a(n)=if(n<0, 0, n++; n*polcoeff(serreverse(x/(1+x+x^3)+x*O(x^n)), n))
%o A127905 (Magma) [0] cat [n*(&+[Binomial(n-1,3*k)*Binomial(3*k,k)/(2*k+1): k in [0..Floor((n-1)/3)]]): n in [1..30]]; // _G. C. Greubel_, Apr 30 2018
%Y A127905 Cf. A005717.
%K A127905 easy,nonn
%O A127905 0,3
%A A127905 _Paul Barry_, Feb 05 2007
%E A127905 Edited by _Charles R Greathouse IV_, Oct 28 2009