A062991 Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).

1, 2, -1, 5, -6, 2, 14, -28, 20, -5, 42, -120, 135, -70, 14, 132, -495, 770, -616, 252, -42, 429, -2002, 4004, -4368, 2730, -924, 132, 1430, -8008, 19656, -27300, 23100, -11880, 3432, -429, 4862, -31824, 92820, -157080, 168300, -116688, 51051, -12870, 1430

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• sign
• easy
• tabl

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

Examples

			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.
		

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.

Crossrefs

Cf. A000108, A002694, A009766, A033184, A062992, A084938, A089434, A094385.

For an unsigned version see Borel's triangle, A234950.

Sums include: A000012 (row), A000079 (diagonal), A064062 (signed row), A071356 (signed diagonal).

Programs

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

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 *)

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

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)