A046521 -id:A046521 - cp's OEIS frontend

A151374 Number of walks within N^2 (the first quadrant of Z^2) starting at (0, 0), ending on the vertical axis and consisting of 2n steps taken from {(-1, -1), (-1, 0), (1, 1)}.

1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, 17199104, 120393728, 852017152, 6085836800, 43818024960, 317680680960, 2317200261120, 16992801914880, 125210119372800, 926554883358720, 6882979133521920, 51309480813527040, 383705682605506560, 2877792619541299200

Offset: 0

Manuel Kauers, Nov 18 2008

A052701 shifted one place left. - R. J. Mathar, Dec 13 2008
Expansion of c(2*x), where c(x) is the g.f. of A000108. - Philippe Deléham, Feb 26 2009; simplified by Alexander Burstein, Jul 31 2018
From Joerg Arndt, Oct 22 2012: (Start)
Also the number of strings of length 2*n of two different types of balanced parentheses.
For example, a(1) = 2, since the two possible strings of length 2 are [] and (), a(2) = 8, since the 8 possible strings of length 4 are (()), [()], ([]), [[]], ()(), [](), ()[], and [][].
The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)
Number of Dyck paths of length 2n in which the step U=(1,1) come in 2 colors. - José Luis Ramírez Ramírez, Jan 31 2013
Row sums of triangle in A085880. - Philippe Deléham, Nov 15 2013
Hankel transform is 2^(n+n^2) = A053763(n+1). - Philippe Deléham, Nov 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
J. Bouttier, P. Di Francesco and E. Guitter, Statistics of planar graphs viewed from a vertex: a study via labeled trees, Nucl. Phys. B, Vol. 675, No. 3 (2003), pp. 631-660. See p. 631, eq. (3.3).
Marek Bożejko, Maciej Dołęga, Wiktor Ejsmont and Światosław R. Gal, Reflection length with two parameters in the asymptotic representation theory of type B/C and applications, arXiv:2104.14530 [math.RT], 2021.
Vedran Čačić and Vjekoslav Kovač, On the fraction of IL formulas that have normal forms, arXiv:1309.3408 [math.LO], 2013.
Stefano Capparelli and Alberto Del Fra, Dyck Paths, Motzkin Paths, and the Binomial Transform, Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.5.
Grégory Chatel and Vincent Pilaud, Cambrian Hopf algebras, Adv. Math. 311, 598-633 (2017). Prop. 3.
Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schröder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016. See eq. (1.13), a=b=2.
Nicolas Crampe, Julien Gaboriaud and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.
Hsien-Kuei Hwang, Mihyun Kang and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
Bradley Robert Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Georg Muntingh, Implicit Divided Differences, Little Schröder Numbers and Catalan Numbers, J. Int. Seq., Vol. 15 (2012), Article 12.6.5; arXiv preprint, arXiv:1204.2709 [math.CO], 2012.
L. Poulain d'Andecy, Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics, arXiv:2304.00850 [math.RT], 2023. See p. 64.

Cf. A000108, A046521, A052701, A053763, A085880.

[2^n * Catalan(n): n in [0..25]]; // Vincenzo Librandi, Oct 24 2012

A151374_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 2*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od;
convert(a,list)end: A151374_list(23); # Peter Luschny, May 19 2011

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]

my(x='x+O('x^66)); Vec(sqrt(2-8*x-2*sqrt(1-8*x))/(4*x)) \\ Joerg Arndt, May 11 2013

def A151374():
    a, n = 1, 1
    while True:
        yield a
        n += 1
        a = a * (8*n - 12) // n
A = A151374()
print([next(A) for  in range(24)]) # _Peter Luschny, Nov 30 2016

a(n) = 2^n * A000108(n). - Philippe Deléham, Feb 01 2009
From Gary W. Adamson, Jul 12 2011: (Start)
a(n) is the top left term in M^n, M = the following infinite square production matrix:
   2, 2, 0, 0, 0, 0, ...
   2, 2, 2, 0, 0, 0, ...
   2, 2, 2, 2, 0, 0, ...
   2, 2, 2, 2, 2, 0, ...
   2, 2, 2, 2, 2, 2, ...
   ...
(End)
E.g.f.: KummerM(1/2, 2, 8*x). - Peter Luschny, Aug 26 2012
From Sergei N. Gladkovskii, Apr 05 2013: (Start)
E.g.f.: Let F(x)=Sum_{n>=0} a(n)*x^n/(2*n)!, then F(x) = E(0)/(1-sqrt(x)) where E(k) = 1 - sqrt(x)/(1 - sqrt(x)/(sqrt(x) - (k+1)*(k+2)/2/E(k+1) )); (continued fraction ).
G.f.: 1 + 4*x/(G(0)-4*x) where G(k) = k*(8*x+1) + 4*x + 2 - 2*x*(2*k+3)*(2*k+4)/G(k+1); (continued fraction). (End)
G.f.: sqrt(2-8*x-2*sqrt(1-8*x))/(4*x). - Mark van Hoeij, May 10 2013
G.f.: (1-sqrt(1-8*x))/(4*x). - Philippe Deléham, Nov 15 2013
D-finite with recurrence (n+1)*a(n) + 4*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Mar 05 2014
a(n) = 4^n*2F1((1-n)/2,-n/2;1;1)/(n+1). - Benedict W. J. Irwin, Jul 12 2016
a(n) ~ 8^n*n^(-3/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016
From Peter Bala, Aug 17 2021: (Start)
a(n) = Sum_{k = 0..floor(n/2)} A046521(n,2*k)*Catalan(2*k).
G.f.: A(x) = 1/sqrt(1 - 4*x)*e(x/(1 - 4*x)), where e(x) = (c(x) + c(-x))/2 is the even part of the function c(x) = (1 - sqrt(1 - 4*x))/(2*x), the g.f. of the Catalan numbers A000108. Inversely, (c(x) + c(-x))/2 = 1/sqrt(1 + 4*x)*A(x/(1 + 4*x)).
x*A(x) = Series reversion of (x - 2*x^2). (End)
Sum_{n>=0} 1/a(n) = 68/49 + 96*arctan(1/sqrt(7)) / (49*sqrt(7)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 20/27 - 16*log(2)/81. - Amiram Eldar, Jan 25 2022
G.f.: 1/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-2*x/(1-...))))))))) (continued fraction). - Nikolaos Pantelidis, Nov 20 2022
a(n) = 2*Sum_{k=1..n} a(k-1)*a(n-k), a(0) = 1. - Mehdi Naima, Jan 16 2023

A048854 Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].

Original entry on oeis.org

1, 2, 1, 12, 12, 1, 120, 180, 30, 1, 1680, 3360, 840, 56, 1, 30240, 75600, 25200, 2520, 90, 1, 665280, 1995840, 831600, 110880, 5940, 132, 1, 17297280, 60540480, 30270240, 5045040, 360360, 12012, 182, 1, 518918400, 2075673600, 1210809600, 242161920, 21621600, 960960, 21840, 240, 1, 17643225600, 79394515200, 52929676800, 12350257920, 1323241920, 73513440, 2227680, 36720, 306, 1

Offset: 0

Wolfdieter Lang

s(n,x) := Sum_{m=0..n} T(n,m)*x^m are monic polynomials satisfying s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} A048786(n,m)*x^m (row polynomials of triangle A048786) and p(0,x)=1.
In the umbral calculus (see the Roman reference, p. 21) the s(n,x) are called Sheffer polynomials for(1/sqrt(1+4*t),t/(1+4*t)). Here the Sheffer notation differs. See the W. Lang link under A006232.
For the general L[d,a] triangles see A286724, also for references.
This is the generalized signless Lah number triangle L[4,1], the Sheffer triangle ((1 - 4*t)^(-1/2), t/(1 - 4*t)). It is defined as transition matrix
risefac[4,1](x, n) = Sum_{m=0..n} L[4,1](n, m)*fallfac[4,1](x, m), where risefac[4,1](x, n) := Product_{0..n-1} (x  + (1 + 4*j)) for n >= 1 and risefac[4,1](x, 0) := 1, and  fallfac[4,1](x, n) := Product_{0..n-1} (x - (1 + 4*j)) for n >= 1 and fallfac[4,1](x, 0) := 1.
In matrix notation: L[4,1] = S1phat[4,1]*S2hat[4,1] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A290319 and A111578 (but here with offsets 0), respectively.
The a- and z-sequences for this Sheffer matrix have e.g.f.s Ea(t) = 1 + 4*t and Ez(t) = (1 + 4*t)*(1 - (1 + 4*t)^(-1/2))/t, respectively. That is, a = {1, 4, repeat(0)} and z(n) = 2*A292220(n). See the W. Lang link on a- and z-sequences there.
The inverse matrix T^(-1) = L^(-1)[4,1] is Sheffer ((1 + 4*t)^(-1/2), t/(1 + 4*t)). This means that T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
fallfac[4,1](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[4,1](x, m), n >= 0.
Diagonal sequences have o.g.f. G(d, x) = A001813(d)*Sum_{m=0..d} A091042(d, m)*x^m/(1 - x)^{2*d + 1}, for d >= 0 (d=0 main diagonal). G(d, x) generates {A001813(d)*binomial(2*(n + d),2*d)}{n >= 0}. See the second W. Lang link on how to compute o.g.f.s of diagonal sequences of general Sheffer triangles. - _Wolfdieter Lang, Oct 12 2017

			The triangle T(n, m) begins:
n\m         0          1          2         3        4      5     6   7 8  ...
0:          1
1:          2          1
2:         12         12          1
3:        120        180         30         1
4:       1680       3360        840        56        1
5:      30240      75600      25200      2520       90      1
6:     665280    1995840     831600    110880     5940    132     1
7:   17297280   60540480   30270240   5045040   360360  12012   182   1
8:  518918400 2075673600 1210809600 242161920 21621600 960960 21840 240 1
...
n = 9: 17643225600 79394515200 52929676800 12350257920 1323241920 73513440 2227680 36720 306 1,
n = 10: 670442572800 3352212864000 2514159648000 670442572800 83805321600 5587021440 211629600 4651200 58140 380 1.
...
Recurrence from a-sequence: T(4, 2) = 2*T(3, 1) + 4*4*T(3, 2) = 2*180 + 16*30 = 840.
Recurrence from z-sequence: T(4, 0) = 4*(z(0)*T(3, 0) + z(1)*T(3, 1) + z(2)*T(3, 2)+ z(3)*T(3, 3)) = 4*(2*120 + 2*180 - 8*30 + 60*1) = 1680.
Four term recurrence: T(4, 2) = T(3, 1) + 2*13*T(3, 2) - 8*3*5*T(2, 2) =  180 + 26*30 - 120*1 = 840.
Meixner type identity for n = 2: (D_x - 4*(D_x)^2)*(12 + 12*x + 1*x^2) = (12 + 2*x) - 4*2 = 2*(2 + x).
Sheffer recurrence for R(3, x): [(2 + x) + 8*(1 + x)*D_x + 16*x*(D_x)^2] (12 + 12*x + 1*x^2) = (2 + x)*(12 + 12*x + x^2) + 8*(1 + x)*(12 + 2*x) + 16*2*x = 120 + 180*x + 30*x^2 + x^3 = R(3, x).
Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (4!*10/2)*(1*30/3! + 4*1/2!) = 840.
Diagonal sequence d = 2: {12, 180, 840 ...} has o.g.f. 12*(1 + 10*x + 5*x^2)/(1 - x)^5  (see A001813(2) and row n=2 of A091042) generating
{12*binomial(2*(n + 2), 4)}_{n >= 0}. - _Wolfdieter Lang_, Oct 12 2017

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Orli Herscovici, Ross G. Pinsky, An identity involving stirling numbers of both kinds and its connection to right-to-left minima of certain set partitions, (2019).
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017, section C) 4.
Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Related to triangle A046521. Cf. A048786. a(n, 0) = A001813.
A111578, A271703 L[1,0], A286724 L[2,1], A290319, A290596 L[3,1], A290597 L[3,2], A292220.
The diagonal sequences are: A000012, 2*A000384(n+1), 12*A053134, 120*A053135, 1680*A053137, ... - Wolfdieter Lang, Oct 12 2017

A290604_row := proc(n) exp(x*t/(1-4*t))/sqrt(1-4*t): series(%, t, n+2): seq(n!*coeff(coeff(%,t,n),x,j), j=0..n) end: seq(A290604_row(n), n=0..9); # Peter Luschny, Sep 23 2017

T[n_, m_] := n!/m! * Binomial[2*n, n] * Binomial[n, m] / Binomial[2*m, m]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)
T[0, 0] = 1; T[-1, ] = T[, -1] = 0; T[n_, m_] /; n < m = 0; T[n_, m_] := T[n, m] = T[n-1, m-1] + 2*(4*n-3)*T[n-1, m] - 8*(n-1)*(2*n-3)*T[n-2, m]; Table[T[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Sep 23 2017 *)

T(n, m) = (n!/m!)*A046521(n, m) = (n!/m!)* binomial(2*n, n)*binomial(n, m)/binomial(2*m, m), n >= m >= 0, a(n, m) := 0, n < m.
Sum_{n>=0, k>=0} T(n, k)*x^n*y^k/(2*n)! = exp(x)*cosh(sqrt(x*y)). - Vladeta Jovovic, Feb 21 2003
T(n, m) = L[4,1](n,m) = Sum_{k=m..n} A290319(n, k)*A111578(k+1, m+1), 0 <= m <= n.
E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
(1 - 4*t)^(-1/2)*exp(x*t/(1 - 4*t)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - 4*t)^(-1/2)*(t/(1 - 4*t))^m/m!, m >= 0.
Three term recurrence for column entries m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 4*n*T(n-1, m) with T(n, m) = 0 for n < m, and for the column m = 0: T(n, 0) = n*Sum_{j=0..n-1} z(j)*T(n-1, j), n >= 1, T(0, 0) = 0, from the a-sequence {1, 4 repeat(0)} and the z(j) = 2*A292220(j) (see above).
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(4*n - 3)*T(n-1, m) - 8*(n-1)*(2*n - 3)*T(n-2, m), n >= m >= 0, with T(0, 0) =1, T(-1, m) = 0, T(n, -1) = 0 and T(n, m) = 0 if n < m.
Meixner type identity for (monic) row polynomials: (D_x/(1 + 4*D_x)) * R(n, x) = n * R(n-1, x), n >= 1, with R(0, x) = 1 and D_x = d/dx. That is, Sum_{k=0..n-1} (-4)^k*(D_x)^(k+1)*R(n, x) = n*R(n-1, x), n >= 1.
General recurrence for Sheffer row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
R(n, x) = [(2 + x)*1 + 8*(1 + x)*D_x + 16*x*(D_x)^2]*R(n-1, x), n >= 1, with R(0, x) = 1.
Boas-Buck recurrence for column m (see a comment in A286724 with references): T(n, m) = (n!/(n-m))*(2 + 4*m)*Sum_{p=0..n-1-m} 4^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, with input T(m, m) = 1.
Explicit form (from the o.g.f.s of diagonal sequences): ((2*(n-m))!/(n-m)!)*binomial(2*n,2*(n-m)), n >= m >= 0, and vanishing for n < m. - Wolfdieter Lang, Oct 12 2017

Name changed, after merging my newer duplicate, from Wolfdieter Lang, Oct 10 2017

A276098 a(n) = (7n)!(3/2n)!/((7n/2)!(3n)!(2n)!).

Original entry on oeis.org

1, 48, 6006, 860160, 130378950, 20392706048, 3254013513660, 526470648692736, 86047769258554950, 14173603389190963200, 2349023203055914140756, 391249767795614684282880, 65434374898388743460014620, 10981406991821583404677201920

Offset: 0

Peter Bala, Aug 22 2016

Let a > b be nonnegative integers. The ratio of factorials (2*a*n)!*(b*n)!/( (a*n)!*(2*b*n)!*((a - b)*n)! ) is known to be an integer for n >= 0 (see, for example, Bober, Theorem 1.1). We have the companion result: Let a > b be nonnegative integers. Then the ratio of factorials ((2*a + 1)*n)!*((b + 1/2)*n)!/(((a + 1/2)*n)!*((2*b + 1)*n)!*((a - b)*n)!) is an integer for n >= 0. This is the case a = 3, b = 1. Other cases include A091496 (a = 2, b = 0), A091527 (a = 1, b = 0), A262732 (a = 2, b = 1), A262733 (a = 3, b = 2) and A276099 (a = 4, b = 2).

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Peter Bala, Some integer ratios of factorials
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A046521, A091496, A091527, A262732, A262733, A276099.

seq(simplify((7*n)!*(3/2*n)!/((7*n/2)!*(3*n)!*(2*n)!)), n = 0..20);

from math import factorial
from sympy import factorial2
def A276098(n): return int((factorial(7*n)*factorial2(3*n)<<(n<<1))//factorial2(7*n)//factorial(3*n)//factorial(n<<1)) # Chai Wah Wu, Aug 10 2023

a(n) = Sum_{k = 0..2*n} binomial(7*n, 2*n - k)*binomial(3*n + k - 1, k).
a(n) = Sum_{k = 0..n} binomial(10*n, 2*n - 2*k)*binomial(3*n + k - 1, k).
Recurrence: a(n) = 28*(7*n - 1)*(7*n - 3)*(7*n - 5)*(7*n - 9)*(7*n - 11)*(7*n - 13)/(3*n*(n - 1)*(2*n - 1)*(2*n - 3)*(3*n - 1)*(3*n - 5)) * a(n-2).
a(n) ~ 1/sqrt(4*Pi*n) * (7^7/3^3)^(n/2).
O.g.f. A(x) = Hypergeom([13/14, 11/14, 9/14, 5/14, 3/14, 1/14], [5/6, 3/4, 1/2, 1/4, 1/6], (7^7/3^3)*x^2) + 48*x*Hypergeom([10/7, 9/7, 8/7, 6/7, 5/7, 4/7], [5/4, 4/3, 3/2, 3/4, 2/3], (7^7/3^3)*x^2).
a(n) = [x^(2*n)] H(x)^n, where H(x) = (1 + x)^7/(1 - x)^3.
It follows that the o.g.f. A(x) equals the diagonal of the bivariate rational generating function 1/2*( 1/(1 - t*H(sqrt(x))) + 1/(1 - t*H(-sqrt(x))) ) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197.
Let F(x) = (1/x)*Series_Reversion( x*sqrt((1 - x)^3/(1 + x)^7) ) and put G(x) = 1 + x*d/dx(log(F(x))). Then A(x^2) = (G(x) + G(-x))/2.

A049444 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -2, 1, 6, -5, 1, -24, 26, -9, 1, 120, -154, 71, -14, 1, -720, 1044, -580, 155, -20, 1, 5040, -8028, 5104, -1665, 295, -27, 1, -40320, 69264, -48860, 18424, -4025, 511, -35, 1, 362880, -663696, 509004, -214676, 54649, -8624, 826, -44, 1, -3628800, 6999840, -5753736

Offset: 0

Wolfdieter Lang

T(n, k) = ^2P_n^k in the notation of the given reference with T(0, 0) := 1. The monic row polynomials s(n,x) := Sum_{m=0..n} T(n, k)*x^k which are s(n, x) = Product_{j=0..n-1} (x-(2+j)), n >= 1 and s(0, x)=1 satisfy s(n, x+y) = Sum_{k=0..n} binomial(n, k)*s(k,x)*S1(n-k, y), with the Stirling1 polynomials S1(n, x) = Sum_{m=1..n} (A008275(n, m)*x^m) and S1(0, x)=1.
In the umbral calculus (see the S. Roman reference given in A048854) the s(n, x) polynomials are called Sheffer polynomials for (exp(2*t), exp(t)-1). This translates to the usual exponential Riordan (Sheffer) notation (1/(1+x)^2, log(1+x)).
See A143491 for the unsigned version of this array and A143494 for the inverse. - Peter Bala, Aug 25 2008
Corresponding to the generalized Stirling number triangle of second kind A137650. - Peter Luschny, Sep 18 2011
Unsigned, reversed rows (cf. A145324, A136124) are the dimensions of the cohomology of a complex manifold with a symmetric group (S_n) action. See p. 17 of the Hyde and Lagarias link. See also the Murri link for an interpretation as the Betti numbers of the moduli space M(0,n) of smooth Riemann surfaces. - Tom Copeland, Dec 09 2016
The row polynomials s(n, x) = (-1)^n*risingfactorial(2 - x, n) are related to the column sequences of the unsigned Abel triangle A137452(n, k), for k >= 2. See the formula there. - Wolfdieter Lang, Nov 21 2022

			The Triangle  begins:
n\k       0       1        2       3       4      5      6    7   8 9 ...
0:        1
1:       -2       1
2:        6      -5        1
3:      -24      26       -9       1
4:      120    -154       71     -14       1
5      -720    1044     -580     155     -20      1
6:     5040   -8028     5104   -1665     295    -27      1
7:   -40320   69264   -48860   18424   -4025    511    -35    1
8:   362880 -663696   509004 -214676   54649  -8624    826  -44
9: -3628800 6999840 -5753736 2655764 -761166 140889 -16884 1266 -54 1
...  [reformatted by _Wolfdieter Lang_, Nov 21 2022]

Y. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Math. Soc. Colloquium Publications Vol. 47, 1999. [From Tom Copeland, Jun 29 2008]
S. Roman, The Umbral Calculus, Academic Press, 1984 (also Dover Publications, 2005).

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces, arXiv:alg-geom/9411004, 1994, (see p. 23, g(x,t)). [From _Tom Copeland_, Dec 11 2011]
T. Hyde and J. Lagarias Polynomial splitting measures and cohomology of the pure braid group, arXiv preprint arXiv:1604.05359 [math.RT], 2016.
Y. Manin, Generating functions in algebraic geometry and sums over trees, arXiv:alg-geom/9407005, 1994, (Eqn. 0.7 and 1.7). [From _Tom Copeland_, Dec 10 2011]
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
R. Murri, Fatgraph Algorithms and the Homology of the Kontsevich Complex, arXiv:1202.1820 [math.AG], 2012, (see Table 1, p. 3). [From _Tom Copeland_, Sep 18 2012]

Unsigned column sequences are A000142(n+1), A001705-A001709. Row sums (signed triangle): n!*(-1)^n, row sums (unsigned triangle): A001710(n-2). Cf. A008275 (Stirling1 triangle).
Cf. A000035, A084938, A094645, A094646.
Cf. A143491, A143494.
Cf. A136124, A137452, A137650.

a049444 n k = a049444_tabl !! n !! k
a049444_row n = a049444_tabl !! n
a049444_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 2)
-- Reinhard Zumkeller, Mar 11 2014

A049444_row := proc(n) local k,i;
add(add(Stirling1(n, n-i), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,k),k=1..n-1) end:
seq(print(A049444_row(n)),n=1..7); # Peter Luschny, Sep 18 2011
A049444:= (n, k)-> add((-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k), j=0..n):
seq(print(seq(A049444(n, k), k=0..n)), n=0..11);  # Mélika Tebni, May 02 2022

t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*(k+1)!*StirlingS1[n-k, i], {k, 0, n-i}]; Flatten[Table[t[n, i], {n, 0, 9}, {i, 0, n}]] [[1 ;; 48]]
(* Jean-François Alcover, Apr 29 2011, after Milan Janjic *)

T(n, k) = T(n-1, k-1) - (n+1)*T(n-1, k), n >= k >= 0; T(n, k) = 0, n < k; T(n, -1) = 0, T(0, 0) = 1.
E.g.f. for k-th column of signed triangle: ((log(1+x))^k)/(k!*(1+x)^2).
Triangle (signed) = [-2, -1, -3, -2, -4, -3, -5, -4, -6, -5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...], where DELTA is Deléham's operator defined in A084938 (unsigned version in A143491).
E.g.f.: (1 + x)^(y-2). - Vladeta Jovovic, May 17 2004 [For row polynomials s(n, y)]
With P(n, t) = Sum_{j=0..n-2} T(n-2,j) * t^j and P(1, t) = -1 and P(0, t) = 1, then G(x, t) = -1 + exp[P(.,t)*x] = [(1+x)^t - 1 - t^2 * x] / [t(t-1)], whose compositional inverse in x about 0 is given in A074060. G(x, 0) = -log(1+x) and G(x, 1) = (1+x) log(1+x) - 2x. G(x, q^2) occurs in formulas on pages 194-196 of the Manin reference. - Tom Copeland, Feb 17 2008
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then T(n,i) = f(n,i,2), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
T(n, k) = Sum_{j=0..n} (-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k). - Mélika Tebni, May 02 2022
From Wolfdieter Lang, Nov 24 2022: (Start)
Recurrence for row polynomials {s(n, x)}_{n>=0}: s(0, x) = 1, s(n, x) = (x - 2)*exp(-(d/dx)) s(n-1, x), for n >= 1. This is adapted from the general Sheffer result given by S. Roman, Corollary 3.7.2., p. 50.
Recurrence for column sequence {T(n, k)}{n>=k}: T(n, n) = 1, T(n, k) = (n!/(n-k))*Sum{j=k..n-1} (1/j!)*(a(n-1-j) + k*beta(n-1-j))*T(n-1, k), for k >= 0, where alpha = repeat(-2, 2) and beta(n) = [x^n] (d/dx)log(log(x)/x) = (-1)^(n+1)*A002208(n+1)/A002209(n+1), for n >= 0. This is the adapted Boas-Buck recurrence, also given in Rainville, Theorem 50., p. 141, For the references and a comment see A046521. (End)

Second formula corrected by Philippe Deléham, Nov 09 2008

A020918 Expansion of 1/(1-4*x)^(7/2).

Original entry on oeis.org

1, 14, 126, 924, 6006, 36036, 204204, 1108536, 5819814, 29745716, 148728580, 730122120, 3528923580, 16830250920, 79342611480, 370265520240, 1712478031110, 7857252142740, 35794148650260

Offset: 0

N. J. A. Sloane

Also convolution of A000984 with A038845, also convolution of A000302 with A002802, also convolution of A002457 with A002697. - Rui Duarte, Oct 08 2011

5*a(n) is the number of (n+3) X 2 Young tableaux with a four horizontal walls between the first and second column. If there is a wall between two cells, the entries may be decreasing; see [Banderier, Wallner 2021], A000984 for one horizontal wall, A002457 for two, and A002802 for three. - Michael Wallner, Mar 09 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Cyril Banderier and Michael Wallner, Young Tableaux with Periodic Walls: Counting with the Density Method, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.

Cf. A000302, A000984, A001622, A002457, A002697, A002802, A000292, A038845, A046521 (fourth column).

List([0..30], n-> Binomial(2*n+6, n+3)*Binomial(n+3, 3)/20); # G. C. Greubel, Jul 20 2019

[Binomial(2*n+6, n+3)*Binomial(n+3, 3)/20: n in [0..30]]; // G. C. Greubel, Jul 20 2019

seq(binomial(2*n,n)*binomial(n,(n-3))/20, n=2..21); # Zerinvary Lajos, May 05 2007
seq(simplify(4^n*hypergeom([-n,-5/2], [1], 1)),n=0..18); # Peter Luschny, Apr 26 2016

CoefficientList[Series[1/(1-4x)^(7/2), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 04 2013 *)

vector(30, n, n--; binomial(2*n+6, n+3)*binomial(n+3, 3)/20 ) \\ G. C. Greubel, Jul 20 2019

[binomial(2*n+6, n+3)*binomial(n+3, 3)/20 for n in (0..30)] # G. C. Greubel, Jul 20 2019

a(n) = binomial(n+3, 3)*A000984(n+3)/A000984(3), where A000984 are the central binomial coefficients. - Wolfdieter Lang
a(n) ~ 8/15*Pi^(-1/2)*n^(5/2)*2^(2*n)*{1 + 35/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
a(n) = Sum_{a+b+c+d+e+f+g=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f)*f(g) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004
a(n) = A000292(n)*A000984(n+2)/20. - Zerinvary Lajos, May 05 2007
From Rui Duarte, Oct 08 2011: (Start)
a(n) = ((2n+5)(2n+3)(2n+1)/(5*3*1)) * binomial(2n, n).
a(n) = binomial(2n+6, 6) * binomial(2n, n) / binomial(n+3, 3).
a(n) = binomial(n+3, 3) * binomial(2n+6, n+3) / binomial(6, 3). (End)
a(n) = 4^n*hypergeom([-n,-5/2], [1], 1). - Peter Luschny, Apr 26 2016
Boas-Buck recurrence: a(n) = (14/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+3, 3). See a comment there. - Wolfdieter Lang, Aug 10 2017
From Amiram Eldar, Apr 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 10*sqrt(3)*Pi - 160/3.
Sum_{n>=0} (-1)^n/a(n) = 10*sqrt(5)*log(phi) - 320/3, where phi is the golden ratio (A001622). (End)
D-finite with recurrence n*a(n) +2*(-2*n-5)*a(n-1)=0. - R. J. Mathar, Aug 01 2022

A158454 Riordan array (1/(1-x^2), x/(1+x)^2).

Original entry on oeis.org

1, 0, 1, 1, -2, 1, 0, 4, -4, 1, 1, -6, 11, -6, 1, 0, 9, -24, 22, -8, 1, 1, -12, 46, -62, 37, -10, 1, 0, 16, -80, 148, -128, 56, -12, 1, 1, -20, 130, -314, 367, -230, 79, -14, 1, 0, 25, -200, 610, -920, 771, -376, 106, -16, 1, 1, -30, 295, -1106, 2083, -2232, 1444, -574, 137, -18, 1

Offset: 0

Paul Barry, Mar 19 2009

Coefficient table of the square of Chebyshev S-polynomials. For the S-polynomials see A049310, and for a proof see the array A181878, where the odd numbered rows are shifted by one to the left. - Wolfdieter Lang, Dec 15 2010
Image of the Catalan numbers A000108 by this matrix is the all 1's sequence.
Image of the central binomial numbers A000984 by this matrix is the counting numbers A000027.
Inverse array is the Riordan array (1-x^2*c(x)^4, xc(x)^2), where c(x) is the g.f. of A000108.
The row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k belong to the class of Boas-Buck polynomials. Hence they satisfy the Boas-Buck identity: (E_x - n*1)*R(n, x) = -Sum_{p=0..n-1} ((1 - (-1)^p)*1 + 2*(-1)^(p+1)*E_x) R(n-1-p, x) for n >= 0. See the Boas-Buck comments and references in A046521. The ensuing recurrence for the column sequences is given in the formula section. - Wolfdieter Lang, Aug 10 2017

			The triangle T(n,k) begins:
  n\k  0   1    2     3     4      5     6     7    8    9  10...
  0:   1
  1:   0   1
  2:   1  -2    1
  3:   0   4   -4     1
  4:   1  -6   11    -6     1
  5:   0   9  -24    22    -8      1
  6:   1 -12   46   -62    37    -10     1
  7:   0  16  -80   148  -128     56   -12     1
  8:   1 -20  130  -314   367   -230    79   -14    1
  9:   0  25 -200   610  -920    771  -376   106  -16    1
  10:  1 -30  295 -1106  2083  -2232  1444  -574  137  -18   1
  ... Reformatted and extended by _Wolfdieter Lang_, Nov 24 2012
Recurrences (from A- and Z-sequences):
  1 = T(6,0) = 0*0 + 1*9 +2*(-24) + 5*22 + 14*(-8)+ 42*1.
-80 = T(7,2) = 1*(-12) -2*(46) -1*(-62) -2*37 -5*(-10) -14*1. - _Wolfdieter Lang_, Dec 20 2010

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.

G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Barry and A. Hennessy, Notes on a Family of Riordan Arrays and Associated Integer Hankel Transforms , JIS 12 (2009) 09.5.3.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 17.
Richard J. Mathar, Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice, viXra:2208.0111, Aug. 2022.

Cf. A000027, A000108, A000984, A006232, A049310, A046521 (Boas-Buck info), A115141, A181878.
From Wolfdieter Lang, Aug 10 2017: (Start)
Row sums A011655(n+1), alternating row sums A007598(n+1)*(-1)^(n+1).
Column sequences k=0..5: A059841, A002620(n+2)*(-1)^(n), A001752(n)*(-1)^n, A001769(n)*(-1)^n, A001780(n)*(-1)^n, A001786(n)*(-1)^n. (End)

T:=Flat(List([0..10], n->List([0..n], k->Sum([0..n], j-> (-1)^(j-k)*Binomial(k+j, 2*k))))); # G. C. Greubel, Dec 15 2018

[[(&+[(-1)^(j-k)*Binomial(k+j, 2*k): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Dec 15 2018

A158454 := proc(n,k) (-1)^(n+k)*add(binomial(n+k-1-2*j,2*k-1),j=0..floor(n/2)) ; end proc;
seq(seq(A158454(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Dec 17 2010

nmax = 10; t[n_, k_] := (-1)^(n+k)* Sum[Binomial[n+k-1-2*j, 2*k-1], {j, 0, Floor[n/2]}]; t[n_?EvenQ, 0] = 1; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 08 2011, after Maple *)
With[{m = 15}, CoefficientList[CoefficientList[Series[(1+x)/((1-x)*(1 + x)^2 -t*x*(1-x)), {x, 0, m}, {t, 0, m}], x], t]]//Flatten (* G. C. Greubel, Dec 15 2018 *)
T[n_, 0] := Boole[EvenQ[n]]; T[n_, k_] := (-1)^(n - k) Binomial[k+n-1, 2*k-1] HypergeometricPFQ[{1, (k - n)/2, (1 + k - n)/2}, {(1 - k - n)/2, (2 - k - n)/2}, 1]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // TableForm  (* Peter Luschny, Aug 20 2022 *)

{T(n,k) = sum(j=0,n, (-1)^(j-k)*binomial(k+j, 2*k))};
for(n=0, 10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 15 2018

[[sum((-1)^(j-k)*binomial(k+j, 2*k) for j in range(n+1)) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Dec 15 2018

Number triangle T(n, k) = Sum_{j=0..n} (-1)^(j-k)*binomial(k+j, 2*k) = Sum_{j=0..n-k} (-1)^(n-k-j)*binomial(n+k-j, 2*k).
O.g.f. column k with leading zeros (Riordan array, see NAME): (1/(1-x^2))*(x/(1+x)^2)^k, k >= 0. - Wolfdieter Lang, Dec 15 2010
T(n, k) = (-1)^(n-k)*Sum_{j=0..floor(n/2)} binomial(n+k-1-2*j, 2*k-1), 0 <= k <= n, else 0. From the o.g.f. for column k after convolution. - Wolfdieter Lang, Dec 17 2010
O.g.f. row polynomials (rising powers in y): ((1+x)/(1-x))/(1+(2-y)*x+x^2) = Sum_{n>=0} S(n,sqrt(y))^2*x^n, with Chebyshev S-polynomials from A049310. - Wolfdieter Lang, Dec 15 2010
Recurrences from the A- and Z-sequences for Riordan arrays. See the W. Lang link under A006232 for details and references.
  T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), n >= 1.
  T(n, k) = Sum_{j=0..n-k} A(j)*T(n-1, k-1+j), n >= k >= 1.
  Here Z(0)=0 and Z(j) = A000108(j), j >= 1, (o.g.f. -1 + c(x), with the Catalan o.g.f. c(x)), and A(j) = A115141(j) = [1,-2,-1,-2,-5,-14,...], j >= 0, with o.g.f. 1/c(x)^2. - Wolfdieter Lang, Dec 20 2010
T(n, k) = Sum_{m=0..n} A129818(m, k), 0 <= k <= n. - Wolfdieter Lang, Dec 15 2010
Boas-Buck recurrence for column k: R(n, k) = (1/(n-k))*Sum_{p=k..n-1}((-1)^(n-p)*(2*k+1) + 1) * R(p, k), for n > k >= 0, with input R(k, k) = 1. See a comment above. - Wolfdieter Lang, Aug 10 2017
G.f.: (1 + x)/((1 - x)*(1 + x)^2 - t*x*(1 - x)). - G. C. Greubel, Dec 15 2018
T(n, k) = (-1)^(n - k)*binomial(k + n - 1, 2*k-1)*hypergeom([1, (k - n)/2, (1 + k - n)/2], [(1 - k - n)/2, (2 - k - n)/2], 1) for k >= 1 . - Peter Luschny, Aug 20 2022

A137452 Triangular array of the coefficients of the sequence of Abel polynomials A(n,x) := x*(x-n)^(n-1).

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 9, -6, 1, 0, -64, 48, -12, 1, 0, 625, -500, 150, -20, 1, 0, -7776, 6480, -2160, 360, -30, 1, 0, 117649, -100842, 36015, -6860, 735, -42, 1, 0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1, 0, 43046721, -38263752, 14880348, -3306744, 459270, -40824, 2268, -72, 1

Offset: 0

Roger L. Bagula, Apr 18 2008

Row sums give A177885.
The Abel polynomials are associated with the Abel operator t*exp(y*t)*p(x) = t*p(x+y).
From Peter Luschny, Jan 14 2009: (Start)
Abs(T(n,k)) is the number of rooted labeled trees on n+1 vertices with a root degree k (Clarke's formula).
The row sums in the unsigned case, Sum_{k=0..n} abs(T(n,k)), count the trees on n+1 labeled nodes, A000272(n+1). (End)
Exponential Riordan array [1, W(x)], W(x) the Lambert W-function. - Paul Barry, Nov 19 2010
The inverse array is the exponential Riordan array [1, x*exp(x)], which is A059297. - Peter Bala, Apr 08 2013
The inverse Bell transform of [1,2,3,...]. See A264428 for the Bell transform and A264429 for the inverse Bell transform. - Peter Luschny, Dec 20 2015
Also the Bell transform of (-1)^n*(n+1)^n. - Peter Luschny, Jan 18 2016

			Triangle begins:
  1;
  0,        1;
  0,       -2,       1;
  0,        9,      -6,       1;
  0,      -64,      48,     -12,      1;
  0,      625,    -500,     150,    -20,      1;
  0,    -7776,    6480,   -2160,    360,    -30,    1;
  0,   117649, -100842,   36015,  -6860,    735,  -42,   1;
  0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1;

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 14 and 29

Seiichi Manyama, Rows n = 0..139, flattened
W. Y. Chen, A general bijective algorithm for trees, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.
L. E. Clarke, On Cayley's formula for counting trees, J. London Math. Soc. 33 (1958), 471-475.
Péter L. Erdős and L. A. Székely, Applications of Antilexicographic Order. I., An Enumerative Theory of Trees, Adv. in Appl. Math. 10, (1989) 488-496.
Eric Weisstein's World of Mathematics, Abel Polynomial.
Wikipedia, Abel Polynomials.
Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

Row sums A177885.
Cf. A000272, A061356, A059297 (inverse array), A264429.
Cf. A061354, A095996, A264234, A061355, A049444.

T := proc(n,k) if n = 0 and k = 0 then 1 else binomial(n-1,k-1)*(-n)^(n-k) fi end; seq(print(seq(T(n,k),k=0..n)),n=0..7); # Peter Luschny, Jan 14 2009
# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-n-1)^n, 9); # Peter Luschny, Jan 27 2016

a0 = 1 a[x, 0] = 1; a[x, 1] = x; a[x_, n_] := x*(x - a0*n)^(n - 1); Table[Expand[a[x, n]], {n, 0, 10}]; a1 = Table[CoefficientList[a[x, n], x], {n, 0, 10}]; Flatten[a1]
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, (-n-1)^n], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

# uses[inverse_bell_transform from A264429]
def A137452_matrix(dim):
    nat = [n for n in (1..dim)]
    return inverse_bell_transform(dim, nat)
A137452_matrix(10) # Peter Luschny, Dec 20 2015

Row n gives the coefficients of the expansion of x*(x-n)^(n-1).
Abs(T(n,k)) = C(n-1,k-1)*n^(n-k). - Peter Luschny, Jan 14 2009
From Wolfdieter Lang, Nov 08 2022: (Start)
From the exponential Riordan (also Sheffer of Jabotinsky) type (1, LambertW) array (see comments).
E.g.f. of column sequence k, LambertW(x)^k/k!, for k >= 0.
E.g.f. of row polynomials P_n(y) = Sum_{k=0..n} T(n, k)*y^k: exp(y*LambertW(x)).
Recurrence for T: T(n, k) = 0 for n < k; T(n, 0) = 1 for n = 0 otherwise 0; T(n, k) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j,k-1)*(-1)^j*T(n-1, k-1+j). (Jabotinsky type convolution triangle, the e.g.f.s for the a- and z-sequences are exp(-x), and 0. See the link in A006232.)
Recurrence for column k of T: T(n, k) = 0 for n < k, T(k, k) = 1, for k >= 0 otherwise T(n, k) = (n!*k/(n-k))*Sum_{j=k..n-1} (1/j!)*beta(n-1-j)*T(j, k), where beta(n) = A264234(n+1)/A095996(n+1) = {-1, 2, -9/2, 32/3, -625/24, ...} with o.g.f. d/dx(log(LambertW(x)/x)). See the Boas-Buck or Rainville references given in A046521, and my Aug 10 2017 comment there.
Recurrence for the row polynomials P_0(x) = 1, and P_n(x) = x*substitute(z=d/dx, exp(-z)/(1+z)) P_(n-1)(x), for n >= 1, with coefficient z^k of exp(-z)/(1+z) given by (-1)^k*A061354(k)/A061355(k). See the Roman reference Corollary 3.7.2., p. 50. (End)
The column sequences for the unsigned triangle Abs(T(n, k)), for k >= 2, are also given by {n^(n-k)*(n-1)*s(k-2, n)/(k-1)!}A049444.%20-%20_Wolfdieter%20Lang">{n>=k} with the row polynomials s(n, x) = risingfactorial(x - (n+1), n) of A049444. - _Wolfdieter Lang, Nov 21 2022

Better name by Peter Bala, Apr 08 2013

Edited by Joerg Arndt, Apr 08 2013

A001769 Expansion of 1/((1+x)*(1-x)^7).

Original entry on oeis.org

1, 6, 22, 62, 148, 314, 610, 1106, 1897, 3108, 4900, 7476, 11088, 16044, 22716, 31548, 43065, 57882, 76714, 100386, 129844, 166166, 210574, 264446, 329329, 406952, 499240, 608328, 736576, 886584, 1061208, 1263576, 1497105, 1765518, 2072862, 2423526, 2822260, 3274194, 3784858

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 17.
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Cf. A002620, A002623, A001752, A001753 (first differences), A158454 (signed column k=3), A001779 (partial sums), A169794 (binomial transf.).

[(4*n^6+96*n^5+910*n^4+4320*n^3+10696*n^2+12864*n+5715)/5760+(-1)^n/128: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

CoefficientList[Series[1/((1+x)(1-x)^7),{x,0,30}],x] (* or *) LinearRecurrence[ {6,-14,14,0,-14,14,-6,1},{1,6,22,62,148,314,610,1106},40] (* Harvey P. Dale, May 24 2015 *)

a(n)=(4*n^6+96*n^5+910*n^4+4320*n^3+10696*n^2+12864*n)\/5760+1 \\ Charles R Greathouse IV, Apr 17 2012

From Paul Barry, Jul 01 2003: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+6, 6).
a(n) = (4*n^6 +96*n^5 +910*n^4 +4320*n^3 +10696*n^2 +12864*n+5715)/5760+(-1)^n/128. (End)
Boas-Buck recurrence: a(n) = (1/n)*Sum_{p=0..n-1} (7 + (-1)^(n-p))*a(p), n >= 1, a(0) = 1. See the Boas-Buck comment in A046521 (here for the unsigned column k = 3 with offset 0).
a(n)+a(n+1) = A000579(n+7). - R. J. Mathar, Jan 06 2021

A068555 Triangle read by rows in which row n contains (2i)!(2j)!/(i!j!*(i+j)!) for i + j = n, i = 0..n.

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 20, 4, 4, 20, 70, 10, 6, 10, 70, 252, 28, 12, 12, 28, 252, 924, 84, 28, 20, 28, 84, 924, 3432, 264, 72, 40, 40, 72, 264, 3432, 12870, 858, 198, 90, 70, 90, 198, 858, 12870, 48620, 2860, 572, 220, 140, 140, 220, 572, 2860, 48620, 184756, 9724

Offset: 0

N. J. A. Sloane, Mar 23 2002

One of three infinite families of integral factorial ratio sequences of height 1 (see Bober, Theorem 1.2). The other two are A007318 and A046521. A related table is A182073. - Peter Bala, Apr 10 2012

			From _Bruno Berselli_, Apr 27 2012: (Start)
Triangle begins:
       1;
       2,    2;
       6,    2,    6;
      20,    4,    4,  20;
      70,   10,    6,  10,  70;
     252,   28,   12,  12,  28, 252;
     924,   84,   28,  20,  28,  84, 924;
    3432,  264,   72,  40,  40,  72, 264, 3432;
   12870,  858,  198,  90,  70,  90, 198,  858, 12870;
   48620, 2860,  572, 220, 140, 140, 220,  572,  2860, 48620;
  184756, 9724, 1716, 572, 308, 252, 308,  572,  1716,  9724, 184756; ...
(End)
T(4,0) = A000984(4) = 70, T(4,1) = 4*20 - 70 = 10, T(4,2) = 4*4 - 10 = 6, T(4,3) = 4*4 - 6 = 10, T(4,4) = 4*20 - 10 = 70. - _Philippe Deléham_, Mar 10 2014

R. K. Guy and Cal Long, Email to N. J. A. Sloane, Feb 22, 2002.
Peter J. Larcombe and David R. French, On the integrality of the Catalan-Larcombe-French sequence 1,8,80,896,10816,.... Proceedings of the Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001). Congr. Numer. 148 (2001), 65-91. MR1887375
Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third edition), page 11.

Vincenzo Librandi, Rows n = 0..100, flattened
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977v1 [math.NT]; J. London Math. Soc. (2) 79 (2009), 422-444.
B. Buca and T. Prosen, Connected correlations, fluctuations and current of magnetization in the steady state of boundary driven XXZ spin chains, arXiv preprint arXiv:1509.04911 [cond-mat.stat-mech], 2015.
Ira Gessel, Rational functions with nonnegative power series, (slides).
Ira Gessel, Super ballot numbers.
Thomas M. Richardson, The Reciprocal Pascal Matrix, arXiv preprint arXiv:1405.6315 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015) # 15.3.3.
Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.
R. Sprugnoli, Riordan array proofs of identities in Gould's book.

Apart perhaps from signs, diagonals give A000984, A002420, A078718.
Cf. A007318, A046521, A182073.
Cf. A182411, A082590 (row sums).

[Factorial(2*i)*Factorial(2*(n-i))/(Factorial(i)*Factorial(n)*Factorial(n-i)): i in [0..n], n in [0..10]]; // Bruno Berselli, Apr 27 2012

A068555 := proc(n,i)
    j := n-i ;
    (2*i)!*(2*j)!/(i!*j!*(i+j)!) ;
end proc: # R. J. Mathar, May 31 2016

Flatten[ Table[ Table[ (2i)!*(2(n - i))!/(i!*(n - i)!*n!), {i, 0, n}], {n, 0, 9}]]

a(n,k)=if(n<0 || k<0,0,(2*n)!*(2*k)!/n!/k!/(n+k)!);

The square array defined by f := (a, b)->add(binomial(2*a, k)*binomial(2*b, a+b-k)*(-1)^(a+b-k), k=0..2*a); and read by antidiagonals gives a signed version. See Sprugnoli, 3.38.
Let f(x) = 1/sqrt(1 - 4*x) denote the o.g.f for A000984. The o.g.f. for this table is (f(x) + f(y))*f(x)*f(y)*(1/(1 + f(x)*f(y))) = (1 + 2*x + 6*x^2 + 20*x^3 + ...) + (2 + 2*x + 4*x^2 + 10*x^3 + ...)*y + (6 + 4*x + 6*x^2 + 12*x^3 + ...)*y^2 + .... - Peter Bala, Apr 10 2012
T(n,0) = A000984(n), T(n,k) = 4*T(n-1,k-1) - T(n,k-1) for k = 1..n. - Philippe Deléham, Mar 10 2014

A104698 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(k, j)*binomial(n-j+1, k+1).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 6, 1, 5, 16, 19, 8, 1, 6, 25, 44, 33, 10, 1, 7, 36, 85, 96, 51, 12, 1, 8, 49, 146, 225, 180, 73, 14, 1, 9, 64, 231, 456, 501, 304, 99, 16, 1, 10, 81, 344, 833, 1182, 985, 476, 129, 18, 1, 11, 100, 489, 1408, 2471, 2668, 1765, 704, 163, 20, 1, 12

Offset: 0

Gary W. Adamson, Mar 19 2005

The n-th column of the triangle is the binomial transform of the n-th row of A081277, followed by zeros. Example: column 3, (1, 6, 19, 44, ...) = binomial transform of row 3 of A081277: (1, 5, 8, 4, 0, 0, 0, ...). A104698 = reversal by rows of A142978. - Gary W. Adamson, Jul 17 2008
This sequence is jointly generated with A210222 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = x*u(n-1,x) + v(n-1) + 1 and v(n,x) = 2x*u(n-1,x) + v(n-1,x) + 1. See the Mathematica section at A210222. - Clark Kimberling, Mar 19 2012
This Riordan triangle T appears in a formula for A001100(n, 0) = A002464(n), for n >= 1. - Wolfdieter Lang, May 13 2025

			The Riordan triangle T begins:
  n\k  0   1   2    3    4    5    6   7   8  9 10 ...
  ----------------------------------------------------
  0:   1
  1:   2   1
  2:   3   4   1
  3:   4   9   6    1
  4:   5  16  19    8    1
  5:   6  25  44   33   10    1
  6:   7  36  85   96   51   12    1
  7:   8  49 146  225  180   73   14   1
  8:   9  64 231  456  501  304   99  16   1
  9:  10  81 344  833 1182  985  476 129  18  1
  10: 11 100 489 1408 2471 2668 1765 704 163 20  1
  ... reformatted and extended by _Wolfdieter Lang_, May 13 2025
From _Wolfdieter Lang_, May 13 2025: (Start)
Zumkeller recurrence (adapted for offset [0,0]): 19 = T(4, 2) = T(2, 1) + T(3, 1) + T(3,3) = 4 + 9 + 6 = 19.
A-sequence recurrence: 19 = T(4, 2) = 1*T(3. 1) + 2*T(3. 2) - 2*T(3, 3) = 9 + 12 - 2 = 19.
Z-sequence recurrence: 5 = T(4, 0) = 2*T(3, 0) - 1*T(3, 1) + 2*T(3, 2) - 6*T(3, 3) = 8 - 9 + 12 + 6 = 5.
Boas-Buck recurrence: 19 = T(4, 2) = (1/2)*((2 + 0)*T(2, 2) + (2 + 2*2)*T(3, 2)) = (1/2)*(2 + 36) = 19. (End)

Reinhard Zumkeller, First 100 rows of triangle, flattened

Diagonal sums are A008937(n+1).
Cf. A048739 (row sums), A008288, A005900 (column 3), A014820 (column 4)
Cf. A081277, A142978 by antidiagonals, A119328, A110271 (matrix inverse).
Cf. A001100, A002464, A010673, A040000, A112478, A133872, A383857.

a104698 n k = a104698_tabl !! (n-1) !! (k-1)
a104698_row n = a104698_tabl !! (n-1)
a104698_tabl = [1] : [2,1] : f [1] [2,1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) ([0] ++ us ++ [0]) $
          zipWith (+) ([1] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Jul 17 2015

A104698 := proc(n, k) add(binomial(k, j)*binomial(n-j+1, n-k-j), j=0..n-k) ; end proc:
seq(seq(A104698(n, k), k=0..n), n=0..15); # R. J. Mathar, Sep 04 2011
T := (n, k) -> binomial(n + 1, k + 1)*hypergeom([-k, k - n], [-n - 1], -1):
for n from 0 to 9 do seq(simplify(T(n, k)), k = 0..n) od;
T := proc(n, k) option remember; if k = 0 then n + 1 elif k = n then 1 else T(n-2, k-1) + T(n-1, k-1) + T(n-1, k) fi end: # Peter Luschny, May 13 2025

u[1, ] = 1; v[1, ] = 1;
u[n_, x_] := u[n, x] = x u[n-1, x] + v[n-1, x] + 1;
v[n_, x_] := v[n, x] = 2 x u[n-1, x] + v[n-1, x] + 1;
Table[CoefficientList[u[n, x], x], {n, 1, 11}] // Flatten (* Jean-François Alcover, Mar 10 2019, after Clark Kimberling *)

T(n,k)=sum(j=0,n-k,binomial(k,j)*binomial(n-j+1,k+1)) \\ Charles R Greathouse IV, Jan 16 2012

The triangle is extracted from the product A * B; A = [1; 1, 1; 1, 1, 1; ...], B = [1; 1, 1; 1, 3, 1; 1, 5, 5, 1; ...] both infinite lower triangular matrices (rest of the terms are zeros). The triangle of matrix B by rows = A008288, Delannoy numbers.
From Paul Barry, Jul 18 2005: (Start)
Riordan array (1/(1-x)^2, x(1+x)/(1-x)) = (1/(1-x), x)*(1/(1-x), x(1+x)/(1-x)).
T(n, k) = Sum_{j=0..n} Sum_{i=0..j-k} C(j-k, i)*C(k, i)*2^i.
T(n, k) = Sum_{j=0..k} Sum_{i=0..n-k-j} (n-k-j-i+1)*C(k, j)*C(k+i-1, i). (End)
T(n, k) = binomial(n+1, k+1)*2F1([-k, k-n], [-n-1], -1) where 2F1 is a Gaussian hypergeometric function. - R. J. Mathar, Sep 04 2011
T(n, k) = T(n-2, k-1) + T(n-1, k-1) + T(n-1, k) for 1 < k < n; T(n, 0) = n + 1; T(n, n) = 1. - Reinhard Zumkeller, Jul 17 2015
From Wolfdieter Lang, May 13 2025: (Start)
The Riordan triangle T = (1/(1 - x)^2, x*(1 + x)/(1 - x)) has the o.g.f. G(x, y) = 1/((1 - x)*(1 - x - y*x*(1+x))) for the row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k.
The o.g.f. for column k is G(k, x) = (1/(1 - x)^2)*(x*(1 + x)/(1 - x))^k, for k >= 0.
The o.g.f. for the diagonal m is D(m, x) = N(m, x)/(1 - x)^(m+1), with the numerator polynomial N(m, x) = Sum_{k=0..floor(m/2)} A034867(m, k)*x^(2*k) for m >= 0.
The row sums with o.g.f. R(x) = 1/((1 -x)*(1 - 2*x -x^2) give A048739.
The alternating row sums with o.g.f. 1/((1 - x)(1 + x^2)) give A133872.
The A-sequence for this Riordan triangle has o.g.f. A(x) = 1 + x + sqrt(1 + 6*x + x^2))/2 giving A112478(n). Hence T(n, k) = Sum_{j=0..n-k} A112478(j)*T(n-1, k-1+j), for n >= 1, k >= 1, T(n, k) = 0 for n < k, and T(0, 0) = 1.
The Z-sequence has o.g.f. (5 + x - sqrt(1 + 6*x + x^2))/2 = 3 + x - A(x) giving Z(n) = {2, -1, -A112478(n >= 2)}. Hence T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), for n >= 1. For A- and Z-sequences of Riordan triangles see a W. Lang link at A006232 with references.
The Boas-Buck sequences alpha and beta for the Riordan triangle T (see A046521 for the Aug 10 2017 comment and reference) are alpha(n) = A040000(n+1) = repeat{2} and beta(n) = A010673(n+1) = repeat{2,0}. Hence the recurrence for column T(n, k){n>=k}, with input T(k, k) = 1, for k >= 0, is T(n, k) = (1/(n-k)) * Sum{j=k..n-1} (2 + k*(1 + (-1)^(n-1-j))) *T(j,k), for n >= k+1. (End)

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A151374 Number of walks within N^2 (the first quadrant of Z^2) starting at (0, 0), ending on the vertical axis and consisting of 2n steps taken from {(-1, -1), (-1, 0), (1, 1)}.

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A048854 Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].

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A276098 a(n) = (7*n)!*(3/2*n)!/((7*n/2)!*(3*n)!*(2*n)!).

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A049444 Generalized Stirling number triangle of first kind.

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A020918 Expansion of 1/(1-4*x)^(7/2).

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A158454 Riordan array (1/(1-x^2), x/(1+x)^2).

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A276098 a(n) = (7n)!(3/2n)!/((7n/2)!(3n)!(2n)!).

A068555 Triangle read by rows in which row n contains (2i)!(2j)!/(i!j!*(i+j)!) for i + j = n, i = 0..n.