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A062991 Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x). 13
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 (list; table; graph; refs; listen; history; text; internal format)
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(a(n,k)*x^k,k=0..n).

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 has T(n,k)=binomial(2n+2,n-k)*binomial(n+k,k)/(n+1). - Paul Barry, May 11 2005

LINKS

Table of n, a(n) for n=0..44.

C. A. Francisco, J. Mermin, 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 ]

A. Lakshminarayan, Z. Puchala, K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169, 2014

FORMULA

a(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.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=0, .., (n-2); a( 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.

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 *)

CROSSREFS

Cf. A000012, A000108, A002694, A009766, A033184, A062992, A084938, A089434.

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

Sequence in context: A231732 A185384 A274728 * A234950 A275228 A118984

Adjacent sequences:  A062988 A062989 A062990 * A062992 A062993 A062994

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang, Jul 12 2001

STATUS

approved

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Last modified February 22 11:50 EST 2020. Contains 332135 sequences. (Running on oeis4.)