A132382 -id:A132382 - cp's OEIS frontend

A132014 T(n,j) for double application of an iterated mixed order Laguerre transform: Coefficients of Laguerre polynomial (-1)^nn!L(n,2-n,x).

1, -2, 1, 2, -4, 1, 0, 6, -6, 1, 0, 0, 12, -8, 1, 0, 0, 0, 20, -10, 1, 0, 0, 0, 0, 30, -12, 1, 0, 0, 0, 0, 0, 42, -14, 1, 0, 0, 0, 0, 0, 0, 56, -16, 1, 0, 0, 0, 0, 0, 0, 0, 72, -18, 1, 0, 0, 0, 0, 0, 0, 0, 0, 90, -20, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, -22, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 132, -24, 1

Offset: 0

Tom Copeland, Oct 30 2007, Nov 05 2007, Nov 11 2007

The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.s A(x) and B(x), or e.g.f.s EA(x) and EB(x).
1) b(0) = a(0), b(1) = a(n) - 2 a(0), b(n) = a(n) - 2n a(n-1) + n(n-1) a(n-2) for n > 0.
2) b(n) = n! Lag{n,(.)!*Lag[.,a1(.),0],-1}, umbrally, where a1(n) = n! Lag{n,(.)!*Lag[.,a(.),0],-1}.
3) b(n) = n! Sum_{j=0..min(2,n)} (-1)^j * binomial(n,j)*a(n-j)/(n-j)!
4) b(n) = (-1)^n n! Lag(n,a(.),2-n)
5) B(x) = (1-xDx)^2 A(x)
6) B(x) = Sum_{j=0..2} {(-1)^j * binomial(2,j)*j!*x^j*Lag(j,-:xD:,0)} A(x)
where D is the derivative w.r.t. x, (:xD:)^j = x^j*D^j and Lag(n,x,m) is the associated Laguerre polynomial of order m.
7) EB(x) = (1-x)^2 EA(x)
8) T = S^2 = A132013^2 = A094587^(-2) = A132159^(-1).
c = (1,-2,2,0,0,...) is the sequence associated to T under the list partition transform and associated operations described in A133314. c are also the coefficients in formula 6. Thus T(n,k) = binomial(n,k)*c(n-k).
The reciprocal sequence to c is d = (1!,2!,3!,4!,...), so the inverse of T is TI(n,k) = binomial(n,k)*d(n-k) = A132159.
These formulas are easily generalized for m applications of the basic operator n! Lag[n,(.)!*Lag[.,a(.),0],-1] by replacing 2 with m in formulas 3, 4, 5, 6 and 7.
The generalized c are given by the generalized coefficients of 6, i.e.,
c(n) = (-1)^n * binomial(m,n)*n! = (-1)^n * m!/(m-n)!.
The generalized d are given by the array at and below the term SI(m-1,m-1) in SI(n,k) = binomial(n,k) * (n-k)!, the inverse of S; i.e.,
d(n) = SI(m-1+n,m-1) = binomial(m-1+n,m-1) * n! = (m-1+n)!/(m-1)!.
As an aside, this shows that the signed falling factorials and the rising factorials form reciprocal pairs under the list partition transform of A133314.
Row sums of T = [formula 3 with all a(n) = 1] = [binomial transform of c] = [coefficients of B(x) with A(x) = 1/(1-x)] = (1,-1,-1,1,5,11,19,...),
with e.g.f. = [EB(x) with EA(x) = exp(x)] = (1-x)^2 * exp(x) = exp(x)*exp(c(.)*x) = exp[(1+c(.))*x].
Alternating row sums of T = [formula 3 with all a(n) = (-1)^n] = [finite differences of c] = [coefficients of B(x) with A(x) = 1/(1+x)] = (1,-3,7,-13,21,-31,...) = (-1)^n A002061(n+1),
with e.g.f. = [EB(x) with EA(x) = exp(-x)] = (1-x)^2 * exp(-x) = exp(- x)*exp(c(.)*x) = exp[-(1-c(.))*x].
See A132159 for a relation to the Poisson-Charlier polynomials. - Tom Copeland, Jan 15 2016

			First few rows of the triangle are
   1;
  -2,   1;
   2,  -4,   1;
   0,   6,  -6,   1;
   0,   0,  12,  -8,   1;
   0,   0,   0,  20, -10,   1;

G. C. Greubel, Rows n=0..100 of triangle, flattened
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Wikipedia, Appell sequence

Cf. A132382, A110330, A132159, A094587, A132013, A235706.

m = 12; s = Exp[x*y]*(1 - x)^2 + O[x]^(m + 2) + O[y]^(m + 2); T[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]*n!; T[0, 0] = 1; Table[T[n, k], {n, 0, m}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2015 *)

row(n) = Vecrev((-1)^n*n!*pollaguerre(n, 2-n)); \\ Michel Marcus, Feb 06 2021

T(n,k) = binomial(n,k)*c(n-k).
E.g.f. for row polynomials: exp(x*y)(1-x)^2. Implies the row polynomials form an Appell sequence (see Wikipedia). - Tom Copeland, Dec 03 2013
From Tom Copeland, Apr 21 2014: (Start)
Change notation letting L(n,m,x) = Lag(n,x,m).
Row polynomials: (-1)^n*n!*L(n,2-n,x) = (-1)^n*(-x)^(n-2)*2!*L(2,n-2,x) =
(-1)^n*(2!/(2-n)!)*K(-n,2-n+1,x) where K is Kummer's confluent hypergeometric function (as a limit of n+s as s tends to zero).
For the row polynomials, the lowering operator = d/dx and the raising operator = x - 2/(1-D).
T = (I - A132440)^2 = [2*I - exp(A238385-I)]^2 = signed exp[2*(A238385-I)], where I = identity matrix.
Operationally, (-1)^n*n!*L(n,2-n,-:xD:) = (-1)^n*x^(n-2)*:Dx:^n*x^(2-n) = (-1)^n*x^(-2)*:xD:^n*x^2 = (-1)^n*n!*binomial(xD+2,n) = (-1)^n*n!*binomial(2,n)*K(-n,2-n+1,-:xD:) where :AB:^n = A^n*B^n for any two operators. Cf. A235706. (End)
n-th row polynomial: n!*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*Lag(k,2,x). - Peter Bala, Jul 25 2021

Title modified by Tom Copeland, Apr 21 2014

Missing term -18 inserted in 10th row by Jean-François Alcover, Jul 09 2015

A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1).

Original entry on oeis.org

1, 2, 1, 0, 4, 1, 0, 0, 6, 1, 0, 0, 0, 8, 1, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 0, 14, 1, 0, 0, 0, 0, 0, 0, 0, 16, 1, 0, 0, 0, 0, 0, 0, 0, 0, 18, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26, 1

Offset: 0

Gary W. Adamson, Oct 07 2007

A134082 * [1,2,3,...] = A084849: (1, 4, 11, 22, 37, ...).
Binomial transform of A134082 = A134083.
A112295 replaces subdiagonal with (-1,-3,-5, ...).

			First few rows of the triangle:
  1;
  2,  1;
  0,  4,  1;
  0,  0,  6,  1;
  0,  0,  0,  8,  1;
  0,  0,  0,  0, 10,  1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A005408, A084849, A112295, A134083.

A134082:= func< n,k | k eq n select 1 else k eq n-1 select 2*n else 0 >;
[A134082(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 17 2021

T[n_, k_]:= If[k==n, 1, If[k==n-1, 2*n, 0]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

def A134082(n,k): return 1 if k==n else 2*n if k==n-1 else 0
flatten([[A134082(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 17 2021

Triangle read by rows, (n-1) zeros followed by (2n, 1). As an infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (2,4,6,8,...) in the subdiagonal.
From formalism in A132382, e.g.f. = I_o[2*(u*x)^(1/2)] (1+2x) where I_o is the zeroth modified Bessel function of the first kind, i.e., I_o[2*(u*x)^(1/2)] = Sum_{j>=0} u^j/j! * x^j/j!. - Tom Copeland, Dec 07 2007
Row polynomial e.g.f.: exp(x*y)(1+2x). - Tom Copeland, Dec 03 2013
Sum_{k=0..n} T(n,k) = 2*n+1 = A005408(n). - G. C. Greubel, Feb 17 2021

More terms added by G. C. Greubel, Feb 17 2021

A055140 Triangle read by rows: T(n, k) = number of matchings of 2n people with partners (of either sex) such that exactly k couples are left together.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 8, 6, 0, 1, 60, 32, 12, 0, 1, 544, 300, 80, 20, 0, 1, 6040, 3264, 900, 160, 30, 0, 1, 79008, 42280, 11424, 2100, 280, 42, 0, 1, 1190672, 632064, 169120, 30464, 4200, 448, 56, 0, 1, 20314880, 10716048, 2844288, 507360, 68544, 7560, 672, 72, 0, 1

Offset: 0

Christian G. Bower, May 09 2000

T is an example of the group of matrices outlined in the table in A132382--the associated matrix for aC(1,1). The e.g.f. for the row polynomials is exp(x*t) * exp(-x) * (1-2*x)^(-1/2). T(n,k) = Binomial(n,k)* s(n-k) where s = A053871 with an e.g.f. of exp(-x) * (1-2*x)^(-1/2) which is the reciprocal of the e.g.f. of A055142. The row polynomials form an Appell sequence. Tom Copeland, Sep 10 2008

A231846 provides a refinement of this array. - Tom Copeland, Oct 12 2016

			Triangle T(n,k) starts:
     1;
     0,    1;
     2,    0,   1;
     8,    6,   0,   1;
    60,   32,  12,   0,  1;
   544,  300,  80,  20,  0, 1;
  6040, 3264, 900, 160, 30, 0, 1;
  ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.

First column is A053871.
Row sums are A001147.
Cf. A055141, A055142.
Cf. A132382, A133314, A231846.

g[0] := 1: g[1] := 0: for n from 2 to 20 do g[n] := (2*(n-1))*(g[n-1]+g[n-2]) end do: T := proc (n, k) options operator, arrow; g[n-k]*binomial(n, k) end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form; Emeric Deutsch, Jan 24 2009

Table[(-1)^# HypergeometricPFQ[{1/2, -#}, {}, 2] Binomial[n, k] &[n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 10 2019, after Eric W. Weisstein at A053871 *)

T(n, k) = A053871(n-k)*binomial(n, k).
From Tom Copeland, Oct 12 2016: (Start)
E.g.f.: e^(xt) e^(-t) (1-2t)^(-1/2) = e^(p.(x)*t)(from my 2008 comment).
Row sums are A001147.
L = D = d/dx and R = x + d[log[e^(L)(1-2L)^(-1/2)]]/dL = x - 1  + 1/(1-2D) = x + 2D + (2D)^2 + (2D)^3 + ... are the lowering and raising operators, i.e., L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x); e.g., L p_2(x) = D (2 + x^2) = 2 x = 2 p_1(x) and R P_2(x) = (x + 2D + 4D^2 + ...) (2 + x^2) = 2x + x^3 + 4x + 8 = 8 + 6x + x^3 = p_3(x).
Another generator is (1-2D)^(-1/2) e^(-D) x^n = (1-2D)^(-1/2) (x-1)^n = p_n(x). For example, (1-2D)^(-1/2)(x-1)^2 = (1 + D + 3 D^2/2 + ...) (x-1)^2 = (x-1)^2 + 2(x-1) + 3 = 2 + x^2 = p_2(x).
Umbral binomial convolution gives p_n(x) = (a. + x)^n = sum_{k = 0,..,n} C(n,k) a_(n-k) * x^k with (a.)^k = a_k = A053871(k).
The Appell sequence of umbral compositional inverses has the e.g.f. e^(xt) e^t (1-2t)^(1/2) associated with A055142. Cf. A231846 for a definition of umbral compositional inversion.
See A132382 and A133314 for more relations.
(End)

A134083 A007318 * A134082.

Original entry on oeis.org

1, 3, 1, 5, 6, 1, 7, 15, 9, 1, 9, 28, 30, 12, 1, 11, 45, 70, 50, 15, 1, 13, 66, 135, 140, 75, 18, 1, 15, 91, 231, 315, 245, 105, 21, 1, 17, 120, 364, 616, 630, 392, 140, 24, 1, 19, 153, 540, 1092, 1386, 1134, 588, 180, 27, 1

Offset: 0

Gary W. Adamson, Oct 07 2007

Row sums = A001787: (1, 4, 12, 32, 80, 192, ...).

A134083 * [1,2,3,...] = A084850: (1, 5, 20, 68, 208, 592, ...).

			First few rows of the triangle:
   1;
   3,  1;
   5,  6,   1;
   7, 15,   9,   1;
   9, 28,  30,  12,   1;
  11, 45,  70,  50,  15,   1;
  13, 66, 135, 140,  75,  18,  1;
  15, 91, 231, 315, 245, 105, 21, 1;
  ...

Cf. A134082, A001787, A084850.

Binomial transform of A134082
From formalism in A132382, e.g.f. = I_o[2*(u*x)^(1/2)] exp(x)(1+2x) where I_o is the zeroth modified Bessel function of the first kind, i.e., I_o[2*(u*x)^(1/2)] = Sum_{j>=0} u^j/j! * x^j/j!. - Tom Copeland, Dec 07 2007
Row polynomial e.g.f.: exp(x*y) * exp(x) * (1+2x). - Tom Copeland, Dec 03 2013

A102625 Triangle read by rows: T(n,k) is the sum of the weights of all vertices labeled k at depth n in the Catalan tree (1 <= k <= n+1, n >= 0).

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 15, 30, 36, 24, 105, 210, 270, 240, 120, 945, 1890, 2520, 2520, 1800, 720, 10395, 20790, 28350, 30240, 25200, 15120, 5040, 135135, 270270, 374220, 415800, 378000, 272160, 141120, 40320, 2027025, 4054050, 5675670, 6486480

Offset: 0

Emeric Deutsch, Jan 31 2005

The Catalan tree is defined as follows: the root is labeled 1 and each vertex labeled i has i+1 children labeled 1,2,...,i+1. The weight of a vertex v is the product of all labels on the path from the root to v. Row n contains n+1 terms. Row sums and column 1 yield the double factorials (A001147). T(n,n+1)=(n+1)!, T(n,n)=n(n+1)!/2 (A001286; Lah numbers).
This table counts permutations of the multiset {1,1,2,2,...,n,n} satisfying the condition "the first appearance of i + 1 follows the first appearance of i" by the position of the first appearance of n. Specifically, T(n+1,k) is the number of such permutations for which n first occurs in position 2n+1-k. For example, with n=2 and k=1, T(3,1)=6 counts 121323, 121332, 122313, 122331, 112323, 112332. - David Callan, Nov 29 2007
T(n+1,k) is also the number of rooted complete binary forests with n labeled leaves and k labeled roots. This follows by comparing exponential generating functions; see Example 5.2.6 and Proposition 5.1.3 of Stanley's "Enumerative Combinatorics 2." - Timothy Y. Chow, Mar 28 2017

			Triangle starts:
   1;
   1,  2;
   3,  6,  6;
  15, 30, 36, 24;
  ...
Production matrix begins:
1, 2
1, 2, 3
1, 2, 3, 4
1, 2, 3, 4, 5
1, 2, 3, 4, 5, 6
1, 2, 3, 4, 5, 6, 7
1, 2, 3, 4, 5, 6, 7, 8
1, 2, 3, 4, 5, 6, 7, 8, 9
... - _Philippe Deléham_, Sep 30 2014
From _Peter Bala_, Apr 16 2017: (Start)
The Catalan tree starts          o1
                                / \
                               /   \
                              /     \
                             /       \
                            /         \
                           o1          o2
                          / \         /|\
                         /   \       / | \
                        /     \     /  |  \
                       o1      o2  o1  o2  o3
Level 2:
2 vertices labeled 1: total weight 1x1x1 + 1x2x1 = 3
2 vertices labeled 2: total weight 2x1x1 + 2x2x1 = 6
1 vertex labeled 3:   total weight 3x2x1         = 6
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
M. R. T. Dale and J. W. Moon, The permuted analogues of three Catalan sets, J. Stat. Plan. Inf. 34 (1) (1993) 75-87 Table 1
S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
Jiaxi Lu and Yuanzhe Ding, A skeleton model to enumerate standard puzzle sequences, arXiv:2106.09471 [math.CO], 2021.
Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

Cf. A001147 (row sums), A001286, A001497, A039683, A132382, A145271, A000108, A033184.

A102625:=proc(n,k) if k<=n+1 then k*(2*n-k+1)!/2^(n-k+1)/(n-k+1)! else 0 fi end proc:
for n from 0 to 8 do seq(A102625(n,k),k=1..n+1) od; # yields sequence in triangular form

t[n_, k_] := k*(2n-k+1)!/(2^(n-k+1)*(n-k+1)!); Table[t[n, k], {n, 0, 8}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Jan 21 2013 *)

{T(n, k) = my(m = n-k+1); if( k<1 || k>n+1, 0, k * (n+m)! / (2^m * m!))}; /* Michael Somos, Aug 16 2016 */

T(n,k) = k*(2*n-k+1)!/[2^(n-k+1)*(n-k+1)!] (1 <= k <= n+1).
From Tom Copeland, Nov 11 2007: (Start)
Bivariate G.F.: exp[P(.,t)*x] = D_x {1 - [g(x)/(1+t*g(x))]} = 1 / {(1+g(x))*[1+t*g(x)]^2}, where g(x) = sqrt(1-2*x) - 1 and P(n,t) = Sum_{k=0..n} T(n,k) * t^k.
Also D_x g(x) = -(1-2*x)^(-1/2) = -exp[x*A001147(.)] = -exp[x *(2*(.)-1)!! ], so the coefficients of x^n/n! in the expansion of g(x) are -(2*(n-1)-1)!! = -A001147(n-1) for n > 0.
See A132382 for an array which is essentially the revert from which this G.f. may be derived and for connections to other arrays. (End)
E.g.f.: 1/(1 - x + x*sqrt(1-2*z)) = 1 + x*z + (x+2*x^2)*z^2/2! + (3*x+6*x^2+6*x^3)*z^3/3! + .... T(n,k) gives the number of plane recursive trees on n+2 nodes where the root has degree k (Bergeron et al., Corollary 5). - Peter Bala, Jul 09 2012
From Peter Bala, Jul 09 2014: (Start)
T(n,k) = k!*A001497(n,k) modulo offset differences.
The n-th row polynomial R(n,x) = (-1)^n/(x - 1)*( Sum_{k = 1..infinity} k*(k - 2)*...*(k - 2*n)*(x/(x - 1))^k ). Cf. the Dobinski-type formula for the row polynomials of A001497. (End)
From Tom Copeland, Aug 06 2016: (Start)
From the 2007 formulas above, an alternate g.f. for this entry is GF(x,t) = -g(x) / [1 + t*g(x)] = x + (1 + 2*t)*x^2/2! + (3 + 6*t + 6*t^2)*x^3/3! + ... with compositional inverse GFinv(x,t) = {1 - [1 - x / (1+t*x)]^2} / 2 = -(1/2)[x / (1+t*x)]^2 + x / (1+t*x) = Sum_{n>0} (-1)^(n+1) [(n-1)/2*t^(n-2) + t^(n-1)]*x^n, a series containing the Lah numbers A001286 when expressed as an e.g.f.
From A145271, with K(x,t) = 1 / dGinv(x,t)/dx = 1 + (1+2*t) x + (1+t+t^2) x^2 + x^3 / [1-(1-t)*x], then [K(x,t) d/dx]^n x evaluated at x=0 gives the n-th row polynomial of this entry.
Since the reciprocal of Bala's e.g.f. above generates a shifted, signed A001147, for the polynomials P(n,t) generated by Bala's e.g.f., umbrally (P(.,t) + a.)^n = 0 for n > 0 with a_0 = 1 and a_n = -t * A001147(n-1) for n > 0. E.g., (P(.,t) + a.)^2 = a_0 * P(2,t) + 2 a_1 * P(1,t) + a_2 * P(0,t) = 1 * (t + 2*t^2) + 2 * -t * t + -t * 1 = 0. (End)
From Peter Bala, Apr 16 2017: (Start)
T(n,k) = k*T(n-1,k-1) + (2*n - k)*T(n-1,k).
E.g.f.: t*x*c(x/2)/(1 - t*x*c(x/2)) = t*x + (t + 2*t^2)*x^2/2! + (3*t + 6*t^2 + 6*t^2)*x^3/3! + ..., where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. for the Catalan numbers A000108. Note that the related g.f. t*x*c(x)/(1 - t*x*c(x)) is the o.g.f. for A033184 (essentially the same as the Riordan array A106566) and enumerates the number of vertices labeled k on the n_th level of the Catalan tree (k >= 1, n >= 0). (End)

A136215 Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 28, 12, 3, 1, 280, 112, 24, 4, 1, 3640, 1400, 280, 40, 5, 1, 58240, 21840, 4200, 560, 60, 6, 1, 1106560, 407680, 76440, 9800, 980, 84, 7, 1, 24344320, 8852480, 1630720, 203840, 19600, 1568, 112, 8, 1, 608608000, 219098880, 39836160

Offset: 0

Paul D. Hanna, Feb 07 2008

Comments from Peter Bala, Jul 10 2008: (Start) This array is the particular case P(1,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown below
n\k|0....................1...............2.........3.....4
----------------------------------------------------------
0..|1.....................................................
1..|a....................1................................
2..|a(a+b)...............2a..............1................
3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
...
See A094587 for some general properties of these arrays.
Other cases recorded in the database include: P(1,0) = Pascal's triangle A007318, P(1,1) = A094587, P(2,0) = A038207, P(3,0) = A027465, P(2,1) = A132159 and P(2,3) = A136216. (End)
The generalized Pascal matrix that Bala refers to is itself a special case of application of the formalism of A133314 to fundamental matrices derived from infinitesimal generators described in A133314, of which the fundamental Pascal (A007318), unsigned Lah (A105278) and associated Laguerre (A135278) matrices are special examples. The formalism gives, among other relations, the inverse of T as TI(n,k) = b(n-k)*C(n,k) where the sequence b is given by the list partition transform (A133314) of A007559; i.e., b = LPT(A007559) = (1,-A008544)= (1,-1,-2,-10,-80,...). The formalism of A132382 may also be applied with the double factorial A001147 replaced by the triple factorial A007559 (see also A133480). - Tom Copeland, Aug 18 2008
From Peter Bala, Aug 29 2013: (Start)
Exponential Riordan array [1/(1 - 3*y)^(1/3), y]. The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x).
For example, row 3 is (28, 12, 3, 1) so P(3,x + 1) = (x + 1)*(x + 4)*(x + 7) = 28 + 12*x + 3*x*(x + 3) + x*(x + 3)*(x + 6). (End)

			Column k of T = column 0 of U^(k+1), while
column k of U = column 0 of T^(3k+1) where U = A136214 and
column k of V = column 0 of T^(3k+2) where V = A112333.
This triangle T begins:
        1;
        1,      1;
        4,      2,     1;
       28,     12,     3,    1;
      280,    112,    24,    4,   1;
     3640,   1400,   280,   40,   5,  1;
    58240,  21840,  4200,  560,  60,  6, 1;
  1106560, 407680, 76440, 9800, 980, 84, 7, 1; ...
Triangle U = A136214 begins:
     1;
     1,    1;
     4,    4,   1;
    28,   28,   7,   1;
   280,  280,  70,  10,  1;
  3640, 3640, 910, 130, 13, 1; ...
with triple factorials A007559 in column 0.
Triangle V = A112333 begins:
      1;
      2,    1;
     10,    5,    1;
     80,   40,    8,   1;
    880,  440,   88,  11,  1;
  12320, 6160, 1232, 154, 14, 1; ...
with triple factorials A008544 in column 0.

G. C. Greubel, Rows n=0..100 of triangle, flattened
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wikipedia, Sheffer sequence

Cf. A136216 (matrix square); A007559, A008544; A136212, A136213.

Cf. A094587.

T[n_, k_]:= Binomial[n, k]*If[n - k == 0, 1, Product[3*j + 1, {j, 0, n - k - 1}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 10 2018 *)

T(n,k)=binomial(n,k)*if(n-k==0,1,prod(j=0,n-k-1,3*j+1))

Column k of T = column 0 of U^(k+1) (matrix power) for k>=0 where U = A136214. Matrix square equals A136216, where A136216(n,k) = A008544(n-k)*C(n,k) where A008544 are also triple factorials.
From Peter Bala, Jul 10 2008: (Start)
T(n,k) = (3*n-3*k-2)*T(n-1,k) + T(n-1,k-1).
E.g.f. exp(x*y)/(1-3*y)^(1/3) = 1 + (1+x)*y + (4+2*x+x^2)*y^2/2! + ... . (End)

A136216 Triangle T, read by rows, where T(n,k) = A008544(n-k)*C(n,k) where A008544 equals the triple factorials in column 0.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 80, 30, 6, 1, 880, 320, 60, 8, 1, 12320, 4400, 800, 100, 10, 1, 209440, 73920, 13200, 1600, 150, 12, 1, 4188800, 1466080, 258720, 30800, 2800, 210, 14, 1, 96342400, 33510400, 5864320, 689920, 61600, 4480, 280, 16, 1

Offset: 0

Paul D. Hanna, Feb 07 2008

This array is the particular case P(2,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown in the comments to A094587. - Peter Bala, Jul 10 2008

The row polynomials form an Appell sequence. - Tom Copeland, Dec 03 2013

			Triangle begins:
1;
2, 1;
10, 4, 1;
80, 30, 6, 1;
880, 320, 60, 8, 1;
12320, 4400, 800, 100, 10, 1;
209440, 73920, 13200, 1600, 150, 12, 1;
4188800, 1466080, 258720, 30800, 2800, 210, 14, 1; ...

Wikipedia, Appell sequence
Wikipedia, Sheffer sequence

Cf. A136215 (square-root), A112333, A008544, A136212, A136213.

Cf. A094587.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 - 3 #)^(2/3)&, #&, 9, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

{T(n,k) = binomial(n,k)*if(n-k==0,1, prod(j=0,n-k-1,3*j+2))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Column k of T = column 0 of V^(k+1) for k>=0 where V = A112333.
Equals the matrix square of triangle A136215.
T(n,k) = (3*n-3*k-1)*T(n-1,k) + T(n-1,k-1). - Peter Bala, Jul 10 2008
Using the formalism of A132382 modified for the triple rather than the double factorial (replace 2 by 3 in basic formulas), the e.g.f. for the row polynomials is exp(x*t)*(1-3x)^(-2/3). - Tom Copeland, Aug 18 2008
From Peter Bala, Aug 28 2013: (Start)
Exponential Riordan array [1/(1 - 3*y)^(2/3), y].
The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (2*x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x). For example, row 3 is (80, 30, 6, 1) so P(3,x + 1) = (2*x + 2)*(2*x + 5)*(2*x + 8) = 80 + 20*(2*x) + 6*(2*x*(2*x + 3)) + (2*x)*(2*x + 3)*(2*x + 6). (End)

A055141 Matrix inverse of triangle A055140.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, -8, -6, 0, 1, -36, -32, -12, 0, 1, -224, -180, -80, -20, 0, 1, -1880, -1344, -540, -160, -30, 0, 1, -19872, -13160, -4704, -1260, -280, -42, 0, 1, -251888, -158976, -52640, -12544, -2520, -448, -56, 0, 1, -3712256, -2266992

Offset: 0

Christian G. Bower, May 09 2000

T is an example of the group of matrices outlined in the table in A132382--the associated matrix for aC(1,1). The e.g.f. for the row polynomials is exp(x*t) * exp(x) * (1-2*x)^(1/2). T(n,k) = Binomial(n,k)* s(n-k) where s = A055142 with an e.g.f. of exp(x) * (1-2*x)^(1/2) which is the reciprocal of the e.g.f. of A053871. The row polynomials form an Appell sequence. [From Tom Copeland, Sep 11 2008]

			1; 0,1; -2,0,1; -8,-6,0,1; -36,-32,-12,0,1; ...

a(n, k) = A053142(n-k)*C(n, k).

A154557 Production array of A122848, read by row.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, -1, 2, 1, 0, 3, -3, 3, 1, 0, -15, 12, -6, 4, 1, 0, 105, -75, 30, -10, 5, 1, 0, -945, 630, -225, 60, -15, 6, 1, 0, 10395, -6615, 2205, -525, 105, -21, 7, 1, 0, -135135, 83160, -26460, 5880, -1050, 168, -28, 8, 1, 0, 2027025, -1216215, 374220, -79380

Offset: 0

Paul Barry, Jan 11 2009

			Array starts
  0, 1,
  0, 1, 1,
  0, -1, 2, 1,
  0, 3, -3, 3, 1,
  0, -15, 12, -6, 4, 1,
  0, 105, -75, 30, -10, 5, 1,
  0, -945, 630, -225, 60, -15, 6, 1,
  0, 10395, -6615, 2205, -525, 105, -21, 7, 1,
  0, -135135, 83160, -26460, 5880, -1050, 168, -28, 8, 1
  0, 2027025, -1216215, 374220, -79380, 13230, -1890, 252, -36, 9, 1
Square of this array is
  0, 1, 1,
  0, 0, 3, 1,
  0, 0, 0, 5, 1,
  0, 0, 0, 0, 7, 1,
  0, 0, 0, 0, 0, 9, 1,
  0, 0, 0, 0, 0, 0, 11, 1,
  0, 0, 0, 0, 0, 0, 0, 13, 1,
  0, 0, 0, 0, 0, 0, 0, 0, 15, 1,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 1

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Cf. A113278, A122848, A132382, A001147.

cp's OEIS Frontend

A132014 T(n,j) for double application of an iterated mixed order Laguerre transform: Coefficients of Laguerre polynomial (-1)^n*n!*L(n,2-n,x).

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A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1).

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A055140 Triangle read by rows: T(n, k) = number of matchings of 2n people with partners (of either sex) such that exactly k couples are left together.

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A134083 A007318 * A134082.

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A102625 Triangle read by rows: T(n,k) is the sum of the weights of all vertices labeled k at depth n in the Catalan tree (1 <= k <= n+1, n >= 0).

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A136215 Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.

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A136216 Triangle T, read by rows, where T(n,k) = A008544(n-k)*C(n,k) where A008544 equals the triple factorials in column 0.

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A132014 T(n,j) for double application of an iterated mixed order Laguerre transform: Coefficients of Laguerre polynomial (-1)^nn!L(n,2-n,x).