A185963 Row sums of number triangle A185962.

1, 0, -2, -3, 0, 7, 11, 1, -24, -40, -7, 82, 145, 37, -279, -524, -174, 945, 1888, 767, -3185, -6783, -3244, 10676, 24301, 13330, -35567, -86823, -53615, 117672, 309366, 212101, -386224, -1099385, -827997, 1255937, 3896480, 3197152, -4039199, -13773374

Offset: 0

Author

Paul Barry, Feb 07 2011

Keywords

• sign
• easy

Examples

			G.f. = 1 - 2*x^2 - 3*x^3 + 7*x^5 + 11*x^6 + x^7 - 24*x^8 - 40*x^9 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
• Index entries for linear recurrences with constant coefficients, signature (2,-3,1).

Crossrefs

Cf. A000931.

Programs

Maple

a := n -> hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], 4/27):
seq(simplify(a(n)), n=0..39); # Peter Luschny, Nov 03 2017

Mathematica

LinearRecurrence[{2,-3,1},{1,0,-2},50] (* Vincenzo Librandi, Feb 18 2012 *)

PARI

x='x+O('x^50); Vec((1-x)^2/(1-2*x+3*x^2-x^3)) \\ G. C. Greubel, Jul 23 2017

Formula

G.f.: (1-x)^2/(1-2x+3x^2-x^3).

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).

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

a(n) = A000931(-3*n). - Michael Somos, Sep 18 2012

a(n) = hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], 4/27). - Peter Luschny, Nov 03 2017

Extensions

More terms from Philippe Deléham, Feb 07 2012