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A185963 Row sums of number triangle A185962.

%I A185963 #42 Jul 09 2025 11:09:51
%S A185963 1,0,-2,-3,0,7,11,1,-24,-40,-7,82,145,37,-279,-524,-174,945,1888,767,
%T A185963 -3185,-6783,-3244,10676,24301,13330,-35567,-86823,-53615,117672,
%U A185963 309366,212101,-386224,-1099385,-827997,1255937,3896480,3197152,-4039199,-13773374
%N A185963 Row sums of number triangle A185962.
%H A185963 Vincenzo Librandi, <a href="/A185963/b185963.txt">Table of n, a(n) for n = 0..1000</a>
%H A185963 Taras Goy and Mark Shattuck, <a href="https://doi.org/10.61091/ojac19-01">Toeplitz-Hessenberg determinant formulas for the sequence F_n-1</a>, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
%H A185963 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,1).
%F A185963 G.f.: (1-x)^2/(1-2x+3x^2-x^3).
%F A185963 a(n) = Sum_{k=0..n} Sum_{i=0..(2k+2)} C(2k+2,i)*Sum_{j=0..(n-k-i)} C(k+j,j)*C(j,n-k-i-j)*(-1)^(n-k-j).
%F A185963 a(n) = Sum_{k=0..n} binomial(n+2k,3k)*(-1)^k = Sum_{k=0..n} A109955(n,k)*(-1)^k. - _Philippe Deléham_, Feb 18 2012
%F A185963 a(n) = A000931(-3*n). - _Michael Somos_, Sep 18 2012
%F A185963 a(n) = hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], 4/27). - _Peter Luschny_, Nov 03 2017
%e A185963 G.f. = 1 - 2*x^2 - 3*x^3 + 7*x^5 + 11*x^6 + x^7 - 24*x^8 - 40*x^9 + ...
%p A185963 a := n -> hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], 4/27):
%p A185963 seq(simplify(a(n)), n=0..39); # _Peter Luschny_, Nov 03 2017
%t A185963 LinearRecurrence[{2,-3,1},{1,0,-2},50] (* _Vincenzo Librandi_, Feb 18 2012 *)
%o A185963 (PARI) x='x+O('x^50); Vec((1-x)^2/(1-2*x+3*x^2-x^3)) \\ _G. C. Greubel_, Jul 23 2017
%Y A185963 Cf. A000931.
%K A185963 sign,easy
%O A185963 0,3
%A A185963 _Paul Barry_, Feb 07 2011
%E A185963 More terms from _Philippe Deléham_, Feb 07 2012