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
Examples
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;
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]
Programs
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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
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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 *)
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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
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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
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)
Comments