A112705 Triangle built from partial sums of Catalan numbers A000108 multiplied by powers.
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 11, 4, 1, 1, 23, 51, 22, 5, 1, 1, 65, 275, 157, 37, 6, 1, 1, 197, 1619, 1291, 357, 56, 7, 1, 1, 626, 10067, 11497, 3941, 681, 79, 8, 1, 1, 2056, 64979, 107725, 46949, 9431, 1159, 106, 9, 1, 1, 6918, 431059, 1045948, 587621, 140681, 19303, 1821, 137, 10, 1
Offset: 0
Examples
Triangle starts: 1; 1, 1; 1, 2, 1; 1, 4, 3, 1; 1, 9, 11, 4, 1; 1, 23, 51, 22, 5, 1; 1, 65, 275, 157, 37, 6, 1; ...
Links
- Wolfdieter Lang, First 10 rows.
Crossrefs
Row sums give A112706.
Programs
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Mathematica
col[m_] := col[m] = CatalanNumber[#]*m^#& /@ Range[0, 20] // Accumulate; T[n_, m_] := If[m == 0, 1, col[m][[n - m + 1]]]; Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Aug 29 2022 *) -
PARI
t(n, m) = if (m==0, 1, if (n==m, 1, sum(kk=0, n-m, m^kk*binomial(2*kk, kk)/(kk+1)))); tabl(nn) = {for (n=0, nn, for (m=0, n, print1(t(n, m), ", ");); print(););} \\ Michel Marcus, Nov 25 2015
Formula
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
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
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