A106268 -id:A106268 - cp's OEIS frontend

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

Paul Barry, Apr 28 2005

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

			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;

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]

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

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

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 *)

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

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

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)

A106269 Expansion of 1/((1 - x^2)*(2 - c(x))), where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 4, 11, 39, 137, 501, 1853, 6936, 26163, 99314, 378879, 1451392, 5579179, 21509692, 83137939, 322049887, 1249941049, 4859617537, 18922572949, 73782881947, 288051510169, 1125832363807, 4404766873969, 17249634205357

Offset: 0

Paul Barry, Apr 28 2005

Row sums of number triangle A106268.
From Petros Hadjicostas, Jul 19 2019: (Start)
Let A(x) be the g.f. of the current sequence. We note first that
Sum_{n >= 3} n*a(n)*x^n = x*A'(x) - (x + 8*x^2),
Sum_{n >= 3} 2*(1-2*n)*a(n-1)*x^n = 2*x*A(x) - 4*x*(x*A(x))' + (2*x + 6*x^2),
Sum_{n >= 3} (-n)*a(n-2)*x^n = -x*(x^2*A(x))' + 2*x^2, and
Sum_{n >= 3} 2*(2*n-1)*a(n-3)*x^n = 4*x*(x^3*A(x))' - 2*x^3*A(x).
Adding these equations (side by side), we get
Sum_{n >= 3} (n*a(n) + 2*(1-2*n)*a(n-1) - n*a(n-2) + 2*(2*n-1)*a(n-3))*x^n = 0,
which proves R. J. Mathar's formula.
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..1664

Cf. A000108, A106268.

A106269:= func< n | (-1)^n*(&+[Binomial(2*k-n, n-2*k): k in [0..Floor(n/2)]]) >;
[A106269(n): n in [0..40]]; // G. C. Greubel, Jan 10 2023

Array[(-1)^#*Sum[Binomial[2 k - #, # - 2 k], {k, 0, Floor[#/2]}] &, 25, 0] (* Michael De Vlieger, Jul 18 2019 *)

c(x) = (1-sqrt(1-4*x))/(2*x);
my(x='x+O('x^35)); Vec(1/((1 - x^2)*(2 - c(x)))) \\ Michel Marcus, Jul 16 2019

def A106269(n): return (-1)^n*sum(binomial(2*k-n, n-2*k) for k in range(n//2+1))
[A106269(n) for n in range(41)] # G. C. Greubel, Jan 10 2023

a(n) = (-1)^n*Sum{k = 0..floor(n/2)} binomial(2*k - n, n - 2*k).
n*a(n) = 2*(2*n-1)*a(n-1) + n*a(n-2) - 2*(2*n-1)*a(n-3). - R. J. Mathar, Dec 10 2011
G.f.: 1/(sqrt(1-4*x)*(1-x^2)*c(x)), where c(x) is the g.f. of A000108. - G. C. Greubel, Jan 10 2023

cp's OEIS Frontend

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

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