A062991 -id:A062991 - cp's OEIS frontend

A062986 Coefficient array for certain polynomials N(5; k,x) (rising powers in x).

1, 5, -10, 10, -5, 1, 35, -170, 415, -629, 630, -420, 180, -45, 5, 285, -2315, 9381, -24395, 44625, -59880, 60015, -45040, 25025, -10010, 2730, -455, 35, 2530, -29379, 169405, -633675, 1703700, -3467145, 5497640, -6903325

Offset: 0

Wolfdieter Lang, Jul 12 2001

The g.f. for the sequence of column r=4*k+j, k >= 0, j=1,2,3,4, of the staircase array A062985(n,r) is N(5; k,x)*(x^(k+1))/(1-x)^(4*k+1+j) with N(5; k,x) := sum(a(k,p)*x^p,p=0..4*k).
The m=0 column gives A002294(k+1). The row sums give A000012 (powers of 1) and (unsigned) A062987.
The sequence of step width of this staircase array is [1,4,4,4,...], i.e. the degree of the row polynomials is [0,4,8,12,...]= A008586.

			{1}; {5,-10,10,-5,1}; {35,-170,415,-629,630,-420,180,-45,5}; ...; N(5; 1,x)= 5-10*x+10*x^2-5*x^3+x^4 = (1-(1-x)^5)/x.

A062991, A062746, A062751.

a(k, p) := [x^p]N(5; k, x) with N(5; k, x)=(N(5; k-1, x)- A002294(k)*(1-x)^(4*k+1))/x, N(5; 0, x) := 1.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=0, .., (4*n-5); a(n, k)= ((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=(4*n-4), ..., 4*n; else 0.

A115126 First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

Original entry on oeis.org

1, 2, 2, 3, 5, 5, 4, 9, 14, 14, 5, 14, 28, 42, 42, 6, 20, 48, 90, 132, 132, 7, 27, 75, 165, 297, 429, 429, 8, 35, 110, 275, 572, 1001, 1430, 1430, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796, 16796, 11, 65, 273, 910

Offset: 1

Wolfdieter Lang, Jan 13 2006

First (k=0) column removed from Catalan triangle A009766(n,k).
In the Derrida et al. 1992 reference this triangle, called here X(alpha=1,beta=1;k=1,n,m), n >= m >= 1, is called there X_{N=n}(K=1,p=m) with alpha=1 and beta=1.
The column sequences give A000027 (natural numbers), A000096, A005586, A005587, A005557, A064059, A064061 for m=1..7. The numerator polynomials for the o.g.f. of column m is found in A062991 and the denominator is (1-x)^(m+1).
The diagonal sequences are convolutions of the Catalan numbers A000108, starting with the main diagonal.

			[1];[2,2];[3,5,5];[4,9,14,14];...
a(4,2) = 9 = binomial(6,2)*3/5.

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.

W. Lang: First 10 rows.

Row sums give A001453(n+1)=A000108(n+1)-1 (Catalan -1).

a(n, m)= binomial(n+m, n)*(n-m+1)/(n+1), n>=m>=1; a(n, m)=0 if n

A157491 A050165*A130595 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 0, -1, 2, 0, 2, -6, 5, 0, -5, 20, -28, 14, 0, 14, -70, 135, -120, 42, 0, -42, 252, -616, 770, -495, 132, 0, 132, -924, 2730, -4368, 4004, -2002, 429, 0, -429, 3432, -11880, 23100, -27300, 19656, -8008, 1430

Offset: 0

Philippe Deléham, Mar 01 2009

Triangle, read by rows, given by [0,-1,-1,-1,-1,-1,-1,...] DELTA [1,1,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938. Triangle related to k-regular trees.

			Triangle begins:
  1;
  0,  1;
  0, -1,  2;
  0,  2, -6,   5;
  0, -5, 20, -28, 14;
  ...

Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A000108, A062991, A094385.

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.

Sum_{k=0..n} T(n,k)*x^(n-k) = A064093, A064092, A064091, A064090, A064089, A064088, A064087, A064063, A064062, A000108, A000012, A064310, A064311, A064325, A064326, A064327, A064328, A064329, A064330, A064331, A064332, A064333 for x = -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 respectively. [Philippe Deléham, Mar 03 2009]

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A062986 Coefficient array for certain polynomials N(5; k,x) (rising powers in x).

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A115126 First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

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A157491 A050165*A130595 as infinite lower triangular matrices.

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