A106270 Inverse of number triangle A106268; triangle T(n,k), 0 <= k <= n.

1, -1, 1, -2, -1, 1, -5, -2, -1, 1, -14, -5, -2, -1, 1, -42, -14, -5, -2, -1, 1, -132, -42, -14, -5, -2, -1, 1, -429, -132, -42, -14, -5, -2, -1, 1, -1430, -429, -132, -42, -14, -5, -2, -1, 1, -4862, -1430, -429, -132, -42, -14, -5, -2, -1, 1, -16796, -4862, -1430, -429, -132, -42, -14, -5, -2, -1, 1

Offset

0,4

Comments

Sequence array for the sequence a(n) = 2*0^n - C(n), where C = A000108 (Catalan numbers). Row sums are A106271. Antidiagonal sums are A106272.

The lower triangular matrix |T| (unsigned case) gives the Riordan matrix R = (c(x), x), a Toeplitz matrix. It is its own so called L-Eigen-matrix (cf. Bernstein - Sloane for such Eigen-sequences, and Barry for such eigentriangles), that is R*R = L*(R - I), with the infinite matrices I (identity) and L with matrix elements L(i, j) = delta(i,j-1) (Kronecker symbol; first upper diagonal with 1s). Thus R = L*(I - R^{-1}), and R^{-1} = I - L^{tr}*R (tr for transposed) is the Riordan matrix (1 - x*c(x), x) given in A343233. (For finite N X N matrices the R^{-1} equation is also valid, but for the other two ones the last row with only zeros has to be omitted.) - Gary W. Adamson and Wolfdieter Lang, Apr 11 2021

Links

Michel Marcus, Table of n, a(n) for n = 0..1325 (Rows n = 0..50 of triangle, flattened).

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490 [math.CO], 2011.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

Formula

Number triangle T(n, k) = 2*0^(n-k) - C(n-k) if k <= n, 0 otherwise; Riordan array (2*sqrt(1-4*x)/(1+sqrt(1-4*x)), x) = (c(x)*sqrt(1-4*x), x), where c(x) is the g.f. of A000108.

Sum_{k=0..n} T(n, k) = A106271(n).

Sum_{k=0..floor(n/2)} T(n, k) = A106272(n).

Bivariate g.f.: Sum_{n, k >= 0} T(n,k)*x^n*y^k = (1/(1 - x*y)) * (2 - c(x)), where c(x) is the g.f. of A000108. - Petros Hadjicostas, Jul 15 2019

From G. C. Greubel, Jan 09 2023: (Start)

Sum_{k=0..n} 2^(n-j)*abs(T(n,k)) = A112696(n).

Sum_{k=0..n} 2^k*abs(T(n,k)) = A014318(n). (End)

Example

Triangle (with rows n >= 0 and columns k >= 0) begins as follows:
     1;
    -1,    1;
    -2,   -1,   1;
    -5,   -2,  -1,   1;
   -14,   -5,  -2,  -1,  1;
   -42,  -14,  -5,  -2, -1,  1;
  -132,  -42, -14,  -5, -2, -1,  1;
  -429, -132, -42, -14, -5, -2, -1,  1;

Mathematica

A106270[n_, k_]:= If[k==n, 1, -CatalanNumber[n-k]];
Table[A106270[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 09 2023 *)

Prog

(PARI) C(n) = binomial(2*n, n)/(n+1); \\ A000108
T(n, k) = if(k <= n, 2*0^(n-k) - C(n-k), 0); \\ Michel Marcus, Nov 11 2022
(Magma)
A106270:= func< n, k | k eq n select 1 else -Catalan(n-k) >;
[A106270(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 09 2023
(SageMath)
def A106270(n, k): return 1 if (k==n) else -catalan_number(n-k)
flatten([[A106270(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 09 2023

Crossrefs

Cf. A000108, A014318, A106268, A106271, A106272, A112696, A343233.

Sequence in context: A174004 A128604 A098885 * A319171 A047888 A330964

Adjacent sequences: A106267 A106268 A106269 * A106271 A106272 A106273

Keyword

easy,sign,tabl

Author

Paul Barry, Apr 28 2005

Status

approved