OFFSET
0,2
COMMENTS
The g.f. for the sequence of column m of triangle A009766(n,m) (or Catalan A033184(n,n-m) diagonals) is N(2; m-1,x)*(x^m)/(1-x)^(m+1), m >= 1, with N(2; n,x) = Sum_{k=0..n} T(n,k)*x^k.
For k=0..1 the column sequences give A000108(n+1) (Catalan), -A002694. The row sums give A000012 (powers of 1) and (unsigned) A062992.
Another version of [1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [0, -1, -1, -1, -1, -1, -1, -1, ...] = 1; 1, 0; 2, -1, 0; 5, -6, 2, 0; 14, -28, 20, -5, 0; 42, -120, 135, -70, 14, 0; ... where DELTA is Deléham's operator defined in A084938.
The positive triangle is |T(n,k)| = binomial(2*n+2, n-k)*binomial(n+k, k)/(n+1). - Paul Barry, May 11 2005
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
C. A. Francisco, J. Mermin, and J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, 2013.
V. E. Hoggatt Jr. and Marjorie Bicknell-Johnson, Numerator Polynomial Coefficient Arrays for Catalan and Related Sequence Convolution Triangles, The Fibonacci Quarterly 15 (1977) 30-34. [On p. 31, in the line n = 1, 14 is missing in S_1^4. - Wolfdieter Lang, Jan 20 2020 ]
Joseph T. Iosue, Adam Ehrenberg, Dominik Hangleiter, Abhinav Deshpande, and Alexey V. Gorshkov, Page curves and typical entanglement in linear optics, arXiv:2209.06838 [quant-ph], 2022.
A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
FORMULA
T(n, k) = [x^k] N(2; n, x) with N(2; n, x) = (N(2; n-1, x) - A000108(n)*(1-x)^(n+1))/x, N(2; 0, x) = 1.
T(n, k) = T(n-1, k+1) + (-1)^k*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=0, .., (n-2); T(n, k) = (-1)^k*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=(n-1) or n; else 0.
O.g.f. (with offset 1) is the series reversion w.r.t. x of x*(1+x*t)/(1+x)^2. If R(n,t) denotes the n-th row polynomial of this triangle then R(n,1-t) is the n-th row polynomial of A009766. Cf. A089434. - Peter Bala, Jul 15 2012
From G. C. Greubel, Sep 27 2024: (Start)
Sum_{k=0..n} T(n, k) = A000012(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A064062(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000079(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A071356(n). (End)
EXAMPLE
The triangle N2 = {a(n,k)} begins:
n\k 0 1 2 3 4 5 6 7 8 9
----------------------------------------------------------------------------
0: 1
1: 2 -1
2: 5 -6 2
3: 14 -28 20 -5
4: 42 -120 135 -70 14
5: 132 -495 770 -616 252 -42
6: 429 -2002 4004 -4368 2730 -924 132
7: 1430 -8008 19656 -27300 23100 -11880 3432 -429
8: 4862 -31824 92820 -157080 168300 -116688 51051 -12870 1430
9: 16796 -125970 426360 -852720 1108536 -969969 570570 -217360 48620 -4862
... formatted by Wolfdieter Lang, Jan 20 2020
N(2; 2, x)= 5 - 6*x + 2*x^2.
MATHEMATICA
T[n_, k_] := 2 (-1)^k Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018 *)
PROG
(Magma)
A062991:= func< n, k | (-1)^k*Binomial(2*n+2, n-k)*Binomial(n+k, k)/(n+1) >;
[A062991(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2024
(SageMath)
def A062991(n, k): return (-1)^k*binomial(2*n+2, n-k)*binomial(n+k, k)/(n+1)
flatten([[A062991(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 27 2024
CROSSREFS
For an unsigned version see Borel's triangle, A234950.
KEYWORD
AUTHOR
Wolfdieter Lang, Jul 12 2001
STATUS
approved