A271703 -id:A271703 - cp's OEIS frontend

A290596 Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].

1, 2, 1, 10, 10, 1, 80, 120, 24, 1, 880, 1760, 528, 44, 1, 12320, 30800, 12320, 1540, 70, 1, 209440, 628320, 314160, 52360, 3570, 102, 1, 4188800, 14660800, 8796480, 1832600, 166600, 7140, 140, 1, 96342400, 385369600, 269758720, 67439680, 7663600, 437920, 12880, 184, 1, 2504902400, 11272060800, 9017648640, 2630147520, 358656480, 25618320, 1004640, 21528, 234, 1, 72642169600, 363210848000, 326889763200, 108963254400, 17335063200, 1485862560, 72836400, 2081040, 33930, 290, 1

Offset: 0

Wolfdieter Lang, Sep 13 2017

For the general L[d,a] triangles see A286724, also for references.
This is the generalized signless Lah number triangle L[3,1], the Sheffer triangle ((1 - 3*t)^(-2/3), t/(1 - 3*t)). It is defined as transition matrix
risefac[3,1](x, n) = Sum_{m=0..n} L[3,1](n, m)*fallfac[3,1](x, m), where risefac[3,1](x, n):= Product_{0..n-1} (x  + (1 + 3*j)) for n >= 1 and risefac[3,1](x, 0) := 1, and  fallfac[3,1](x, n):= Product_{0..n-1} (x - (1 + 3*j)) for n >= 1 and fallfac[3,1](x, 0) := 1.
In matrix notation: L[3,1] = S1phat[3,1]*S2hat[3,1] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A286718 and A111577 (but here with offsets 0), respectively.
The a- and z-sequences for this Sheffer matrix has e.g.f.s Ea(t) = 1 + 3*t and (Ez(t) = (1 + 3*t)*(1 - (1 + 3*t)^(-2/3))/t, respectively. That is, a = {1, 3, repeat(0)} and z(n) = A290597(n)/A038500(n+1). For the proof see the second W. Lang link. See also a W. Lang link under A006232 for Sheffer a- and z-sequences with references (in the Riordan case).
The inverse matrix T^(-1) = L^(-1)[3,1] is Sheffer ((1 + 3*t)^(-2/3), t/(1 + 3*t)). This means that  T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
fallfac[3,1](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[3,1](x, m), n >= 0.

			The triangle T(n, m) begins:
n\m         0         1         2        3       4      5     6   7 8  ...
0:          1
1:          2         1
2:         10        10         1
3:         80       120        24        1
4:        880      1760       528       44       1
5:      12320     30800     12320     1540      70      1
6:     209440    628320    314160    52360    3570    102     1
7:    4188800  14660800   8796480  1832600  166600   7140   140   1
8:   96342400 385369600 269758720 67439680 7663600 437920 12880 184 1
...
n = 9: 2504902400 11272060800 9017648640 2630147520 358656480 25618320 1004640 21528 234 1,
n = 10: 72642169600 363210848000 326889763200 108963254400 17335063200 1485862560 72836400 2081040 33930 290 1.
...
Recurrence from a-sequence:  T(4, 2) = 2*T(3, 1) + 3*4*T(3, 2) = 2*120 + 12*24 = 528.
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*80 + 1*120 - (10/3)*24 + 20*1) = 880.
Four term recurrence: T(4, 2) = T(3, 1) + 2*10*T(3, 2) - 3*3*8*T(2, 2) =  120 + 20*24 - 72*1 = 528.
Meixner type identity for n = 2: (D_x - 3*(D_x)^2)*(10 + 10*x + x^2 ) = (10 + 2*x) - 3*2 = 2*(2 + x).
Sheffer recurrence for R(3, x): [(2 + x) + 6*(1 + x)*D_x + 9*x*(D_x)^2] (10 + 10*x + x^2) = (2 + x)*(10 + 10*x + x^2) + 6*(1 + x)*(10 +2*x) + 9*2*x = 80 + 120*x + 24*x^2 + x^3 = R(3, x).
Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (4!*8/2)*(1*24/3! + 3*1/2!) = 528.

Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.
Wolfdieter Lang, Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.

Cf. A008544 (column m=0), A038500, A111577, A271703 L[1,0], A286718, A286724 L[2,1], A290597, A290598 L[3,2].

T(n, m) = L[3,1](n,m) = Sum_{k=m..n} A286718(n, k)*A111577(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 - 3*t)^(-2/3)*exp(x*t/(1 - 3*t)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - 3*t)^(-2/3)*(t/(1 - 3*t))^m/m!, m >= 0.
Three term recurrence for column entries m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 3*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), from the a-sequence {1, 3 repeat(0)} and the z-sequence given above.
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(3*n - 2)*T(n-1, m) - 3*(n-1)*(3*n - 4)*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 + 3*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} (-3)^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 + 6*(1 + x)*D_x + 3^2*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 + 3*m)*Sum_{p=0..n-1-m} 3^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, with input T(m, m) = 1.

A134432 Sum of entries in all the arrangements of the set {1,2,...,n} (to n=0 there corresponds the empty set).

Original entry on oeis.org

0, 1, 9, 66, 490, 3915, 34251, 328804, 3452436, 39456405, 488273005, 6510306726, 93097386174, 1421850988831, 23105078568495, 398118276872520, 7251440043035176, 139227648826275369, 2810658160680434001, 59519819873232720010, 1319356007189991960210

Offset: 0

Emeric Deutsch, Nov 16 2007

nonn

Appears to be the binomial transform of A001286 (filled with the appropriate two leading zeros), shifted one index left. - R. J. Mathar, Apr 04 2012

			a(2)=9 because the arrangements of {1,2} are (empty), 1, 2, 12 and 21.

Alois P. Heinz, Table of n, a(n) for n = 0..447

Cf. A000522, A001286, A134431, A271703, A271705, A326659.

[Binomial(n+1,2)*(&+[Factorial(j)*Binomial(n-1, j-1): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 09 2022

Q[0]:=1: for n to 17 do Q[n]:=sort(simplify(Q[n-1]+t^n*x*(diff(x*Q[n-1], x))), t) end do: for n from 0 to 17 do P[n]:=sort(subs(x=1,Q[n])) end do: seq(subs(t =1,diff(P[n],t)),n=0..17);
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, [t!, 0],
      b(n-1, t)+(p-> p+[0, n*p[1]])(b(n-1, t+1)))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 19 2020

(* First program *)
b[n_, s_, t_]:= b[n, s, t] = If[n==0, t! x^s, b[n-1, s, t] + b[n-1, s+n, t+1]];
T[n_]:= T[n]= Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]] @ b[n, 0, 0];
a[n_] := Sum[k T[n][[k+1]], {k, 0, n(n+1)/2}];
a /@ Range[0, 20] (* Jean-François Alcover, Feb 19 2020, after Alois P. Heinz *)
(* Second program *)
a[n_]:= ((n+1)/2)*Sum[j*j!*Binomial[n,j], {j,0,n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 09 2022 *)

[((n+1)/2)*sum( j*factorial(j)*binomial(n, j) for j in (0..n) ) for n in (0..30)] # G. C. Greubel, Jan 09 2022

a(n) = Sum_{k=0..n*(n+1)/2} k*A134431(n,k).
a(n) = (d/dt)P[n](t) evaluated at t=1; here P[n](t)=Q[n](t,1) where the polynomials Q[n](t,x) are defined by Q[0]=1 and Q[n]=Q[n-1] + xt^n (d/dx)xQ[n-1]. (Q[n](t,x) is the bivariate generating polynomial of the arrangements of {1,2,...,n}, where t (x) marks the sum (number) of the entries; for example, Q[2](t,x) = 1 + tx + t^2*x + 2t^3*x^2, corresponding to: empty, 1, 2, 12 and 21, respectively.)
E.g.f.: exp(x)*x*(2 + x - x^2) / (2*(1 - x)^3). - Ilya Gutkovskiy, Jun 02 2020
From G. C. Greubel, Jan 09 2022: (Start)
a(n) = A271705(n+1, 2).
a(n) = ((n+1)/2) * Sum_{j=0..n} j * j! * binomial(n, j).
a(n) = (1/n!)*binomial(n+1, 2) * Sum_{j=0..n} (j!)^2 * A271703(n, j). (End)
D-finite with recurrence (-n+1)*a(n) +(n+1)^2*a(n-1) -n*(n+1)*a(n-2)=0. - R. J. Mathar, Jul 26 2022

More terms from Alois P. Heinz, Dec 22 2017

A292219 Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].

Original entry on oeis.org

1, 6, 1, 60, 20, 1, 840, 420, 42, 1, 15120, 10080, 1512, 72, 1, 332640, 277200, 55440, 3960, 110, 1, 8648640, 8648640, 2162160, 205920, 8580, 156, 1, 259459200, 302702400, 90810720, 10810800, 600600, 16380, 210, 1, 8821612800, 11762150400, 4116752640, 588107520, 40840800, 1485120, 28560, 272, 1

Offset: 0

Wolfdieter Lang, Sep 23 2017

For the general L[d,a] triangles see A286724, also for references.
This is the generalized signless Lah number triangle L[4,3], the Sheffer triangle ((1 - 4*t)^(-3/2), t/(1 - 4*t)). It is defined as transition matrix
risefac[4,3](x, n) = Sum_{m=0..n} L[4,3](n, m)*fallfac[4,3](x, m), where risefac[4,3](x, n) := Product_{0..n-1} (x  + (3 + 4*j)) for n >= 1 and risefac[4,3](x, 0) := 1, and fallfac[4,3](x, n):= Product_{0..n-1} (x - (3 + 4*j)) for n >= 1 and fallfac[4,3](x, 0) := 1.
In matrix notation: L[4,3] = S1phat[4,3]*S2hat[4,3] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A225471 and A225469, 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)^(-3/2))/t, respectively. That is, a = {1, 4, repeat(0)} and z(n) = 2*A292221(n). See the W. Lang link on a- and z-sequences there.
The inverse matrix T^(-1) = L^(-1)[4,3] is Sheffer ((1 + 4*t)^(-3/2), t/(1 + 4*t)). This means that T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
  fallfac[4,3](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[4,3](x, m), n >= 0.
Diagonal sequences have o.g.f. G(d, x) = A001813(d)*Sum_{m=0..d} A103327(d, m)*x^m/(1 - x)^(2*d + 1), for d >= 0 (d=0 main diagonal). G(d, x) generates {A001813(d)*binomial(2*(m + d) + 1, 2*d)}{m >= 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:           6           1
  2:          60          20          1
  3:         840         420         42         1
  4:       15120       10080       1512        72        1
  5:      332640      277200      55440      3960      110       1
  6:     8648640     8648640    2162160    205920     8580     156     1
  7:   259459200   302702400   90810720  10810800   600600   16380   210   1
  8:  8821612800 11762150400 4116752640 588107520 40840800 1485120 28560 272  1
  ...
Recurrence from a-sequence: T(4, 2) = (4/2)*T(3, 1) + 4*4*T(3, 2) = 2*420 + 16*42 = 1512.
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*(6*840 - 6*420 + 40*42 -420*1) = 15120.
Meixner type identity for n = 2: (D_x - 4*(D_x)^2)*(60 + 20*x + 1*x^2 ) = (20 + 2*x) - 4*2 = 2*(6 + x).
Sheffer recurrence for R(3, x): [(6 + x) + 8*(3 + x)*D_x + 16*x*(D_x)^2] (60 + 20*x + 1*x^2) = (6 + x)*(60 + 20*x + x^2) + 8*(3 + x)*(20 + 2*x) + 16*2*x = 840 + 420*x + 42*x^2 + x^3 = R(3, x).
Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (2*4!/2)*(3 + 2*2)*(1*42/3! + 4*1/2!) = 1512.
Diagonal sequence d = 2: {60, 420, 1512, ...} has o.g.f. 12*(5 + 10*x + x^2)/(1 - x)^5 (see A001813(2) and row n=2 of A103327) generating {12*binomial(2*(m + 2) + 1, 4)}_{m >= 0}. - _Wolfdieter Lang_, Oct 12 2017

Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.

Wolfdieter Lang, Table of n, a(n) for n = 0..1325
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.

Cf. A225469, A225471, A271703 L[1,0], A286724 L[2,1], A290596 L[3,1], A290597 L[3,2], A048854 L[4,1], A292221, A103327,

Diagonal sequences: A000012, 2*A014105(m+1), 12*A053126(m+4), 120*A053128(m+6), A053130(n+8), ... - Wolfdieter Lang, Oct 12 2017

T(n, m) = L[4,3](n,m) = Sum_{k=m..n} A225471(n, k)*A225469(k, m), 0 <= m <= n.
E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
(1 - 4*t)^(-3/2)*exp(x*t/(1 - 4*t)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - 4*t)^(-3/2)*(t/(1 - 4*t))^m/m!, m >= 0.
Three term recurrence for column entries k >= 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 z(j) = 2*A292221(j) (see above).
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(4*n - 1)*T(n-1, m) - 8*(n-1)*(2*n - 1)*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) = [(6 + x)*1 + 8*(3 + 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) = (2*n!/(n-m))*(3 + 2*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 + 1, 2*(n-m)), n >= m >= 0, and vanishing for n < m. - Wolfdieter Lang, Oct 12 2017

A290598 Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,2].

Original entry on oeis.org

1, 4, 1, 28, 14, 1, 280, 210, 30, 1, 3640, 3640, 780, 52, 1, 58240, 72800, 20800, 2080, 80, 1, 1106560, 1659840, 592800, 79040, 4560, 114, 1, 24344320, 42602560, 18258240, 3043040, 234080, 8778, 154, 1, 608608000, 1217216000, 608608000, 121721600, 11704000, 585200, 15400, 200, 1, 17041024000, 38342304000, 21909888000, 5112307200, 589881600, 36867600, 1293600, 25200, 252, 1, 528271744000, 1320679360000, 849008160000, 226402176000, 30477216000, 2285791200, 100254000, 2604000, 39060, 310, 1

Offset: 0

Wolfdieter Lang, Sep 13 2017

For the general L[d,a] triangles see A286724, also for references.
This is the generalized signless Lah number triangle L[3,2], the Sheffer triangle ((1 - 3*t)^(-4/3), t/(1 - 3*t)). It is defined as transition matrix risefac[3,2](x, n) = Sum_{m=0..n} L[3,2](n, m)*fallfac[3,2](x, m), where risefac[3,2](x, n):= Product_{0..n-1} (x  + (2 + 3*j)) for n >= 1 and risefac[3,2](x, 0) := 1, and  fallfac[3,2](x, n):= Product_{0..n-1} (x - (2 + 3*j)) for n >= 1 and fallfac[3,2](x, 0) := 1.
In matrix notation: L[3,2] = S1phat[3,2]*S2hat[3,2] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A225470 and A225468, respectively.
The a- and z-sequences for this Sheffer matrix have e.g.f.s 1 + 3*t and (1 + 3*t)*(1 - (1 + 3*t)^(-4/3))/t, respectively. That is, a = {1, 3, repeat(0)} and z(n) = A290603(n)/A038500(n+1). See a W. Lang link under A006232 for these types of sequences with a reference, and also the present link, eq. (142).
The inverse matrix T^(-1) = L^(-1)[3,2] is Sheffer ((1 + 3*t)^(-4/3), t/(1 + 3*t)). This means that T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
fallfac[3,2](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[3,2](x, m), n >= 0.

			The triangle T(n, m) begins:
n\m          0          1         2         3        4      5     6   7 8  ...
0:           1
1:           4          1
2:          28         14         1
3:         280        210        30         1
4:        3640       3640       780        52        1
5:       58240      72800     20800      2080       80      1
6:     1106560    1659840    592800     79040     4560    114     1
7:    24344320   42602560  18258240   3043040   234080   8778   154   1
8:   608608000 1217216000 608608000 121721600 11704000 585200 15400 200 1
...
n = 9: 17041024000 38342304000 21909888000 5112307200 589881600 36867600 1293600 25200 252 1,
n = 10: 528271744000 1320679360000 849008160000 226402176000 30477216000 2285791200 100254000 2604000 39060 310 1.
...
Recurrence from a-sequence:  T(4, 2) = (4/2)*T(3, 1) + 3*4*T(3, 2) = 2*210 + 12*30 = 780.
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*(4* 280  - 2*210 + (28/3)*30 - 70*1) = 3640.
Four term recurrence: T(4, 2) = T(3, 1) + 2*11*T(3, 2) - 3*3*10*T(2, 2) =  210 + 22*30  - 90*1 = 780.
Meixner type identity for n = 2: (D_x - 3*(D_x)^2)*(28 + 14*x + x^2) = (14 + 2*x) - 3*2 = 2*(4 + x).
Sheffer recurrence for R(3, x): [(4 + x) + 6*(2 + x)*D_x + 9*x*(D_x)^2] (28 + 14*x + x^2) = (4 + x)*(28 + 14*x + x^2) + 6*(2 + x)*(14 + 2*x) + 9*2*x= 280 + 210*x + 30*x^2 + x^3 =  R(3, x).
Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (4!*(4 + 3*2)/2)*(1*30/3! + 3*1/2!) = 780.

Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.

Cf. A007559(n+1) (column m=0), A225468, A225470, A271703 L[1,0], A286724 L[2,1], A290596, L[3,1], A290603.

T(n, m) = L[3,2](n,m) = Sum_{k=m..n} A225470(n, k) * A225468(k, m), 0 <= m <= n.
E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
(1 - 3*t)^(-4/3)*exp(x*t/(1 - 3*t)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - 3*t)^(-4/3)*(t/(1 - 3*t))^m/m!, m >= 0.
Three term recurrence for column entries m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 3*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), from the a-sequence {1, 3 repeat(0)} and the z-sequence given above.
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(3*n - 1)*T(n-1, m) - 3*(n-1)*(3*n - 2)*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 + 3*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} (-3)^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) = [(4 + x)*1 + 6*(2 + x)*D_x + 3^2*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))*(4 + 3*m)*Sum_{p=0..n-1-m} 3^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, with input T(m, m) = 1.

A349776 Triangle read by rows: T(n,k) is the number of partitions of set [n] into a set of at most k lists, with 0 <= k <= n. Also called broken permutations.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 6, 12, 13, 0, 24, 60, 72, 73, 0, 120, 360, 480, 500, 501, 0, 720, 2520, 3720, 4020, 4050, 4051, 0, 5040, 20160, 32760, 36960, 37590, 37632, 37633, 0, 40320, 181440, 322560, 381360, 393120, 394296, 394352, 394353

Offset: 0

Ron L.J. van den Burg, Nov 29 2021

List means an ordered subset.

			For n=3 the T(3, 2)=12 broken permutations are {(1, 2, 3)}, {(1, 3, 2)}, {(2, 1, 3)}, {(2, 3, 1)}, {(3, 1, 2)}, {(3, 2, 1)}, {(1, 2), (3)}, {(2, 1), (3)}, {(1, 3), (2)}, {(3, 1), (2)}, {(2, 3), (1)}, {(3, 2), (1)}.
If you add the set of 3 lists {(1), (2), (3)}, you get T(3, 3) = 13 = A000262(3).
Triangle begins:
  1;
  0,   1;
  0,   2,    3;
  0,   6,   12,   13;
  0,  24,   60,   72,   73;
  0, 120,  360,  480,  500,  501;
  0, 720, 2520, 3720, 4020, 4050, 4051;
  ...

Kenneth P. Bogart, Combinatorics Through Guided Discovery, Kenneth P. Bogart, 2004, 57-58.

David Callan, Sets, Lists and Noncrossing Partitions, arXiv:0711.4841 [math.CO], 2007-2008. Also Sets, Lists and Noncrossing Partitions, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.3.
Wikipedia, The twentyfold way

Columns k=0-2 give (for n>=k): A000007, A000142, A001710.
Partial sums of A271703 per row.
Main diagonal is A000262.
Row sums give A062147(n-1) for n>=1.
Cf. A096965.

T:= proc(n, k) option remember; `if`(k<0, 0,
      binomial(n-1, k-1)*n!/k! +T(n, k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 01 2021

T[n_, k_] := Sum[Binomial[n-1, n-j] n!/j!, {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2023 *)

Lah(n, k) = if (n==0, 1, binomial(n-1, k-1)*n!/k!); \\ A271703
T(n, k) = sum(j=0, k, Lah(n, j)); \\ Michel Marcus, Nov 30 2021

def T(n, k):
    return sum(binomial(n, i)*falling_factorial(n-1, n-i) for i in (0..k))
print([[T(n, k) for k in (0..n)] for n in (0..9)])  # Peter Luschny, Dec 01 2021

T(n, k) = Sum_{j=0..k} A271703(n, j) for n >= 0.
T(n, n) = A000262(n).
T(n, k) = Sum_{j=0..k} binomial(n-1, n-j)*n!/j!.
T(n, k) = A000262(n) - A271703(n, k + 1) * hypergeom([1, k - n + 1], [k + 1, k + 2], -1). - Peter Luschny, Nov 30 2021
|Sum_{k=0..n} (-1)^k * T(n,k)| = A096965(n). - Alois P. Heinz, Dec 01 2021

A357367 Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.

Original entry on oeis.org

1, 0, 2, 0, 6, 12, 0, 24, 120, 120, 0, 120, 1080, 2520, 1680, 0, 720, 10080, 40320, 60480, 30240, 0, 5040, 100800, 604800, 1512000, 1663200, 665280, 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280

Offset: 0

Peter Luschny, Sep 26 2022

T(n, k) is the cardinality of the set of all phylogenetic trees with linearly ordered children having n + 1 leaves and k internal vertices. (Proposition 4.16 in Deb and Sokal). - Peter Luschny, Aug 06 2025

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0,     2;
  [2] 0,     6,      12;
  [3] 0,    24,     120,     120;
  [4] 0,   120,    1080,    2520,     1680;
  [5] 0,   720,   10080,   40320,    60480,    30240;
  [6] 0,  5040,  100800,  604800,  1512000,  1663200,   665280;
  [7] 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened).
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 33.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See pp. 3, 5.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Cf. A032037 (row sums), A271703, A386789.

T := (n, k) -> add((-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!, m = 0..k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
T := proc(n, k) option remember; if n = 0 and k = 0 then 1 elif k <= 0 or n < 0 then 0 else 2*(n + k - 1)*T(n-1, k-1) + (n + 2*k - 1)*T(n-1, k) fi end:
for n from 0 to 6 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Aug 06 2025

T[n_, k_] := Sum[(-1)^(m + k)*Binomial[n + k, n + m]*Binomial[n + m - 1, m - 1]*(n + m)!/m!, {m, 0, k}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 05 2025 *)

def Lah(n, k): return binomial(n, k) * falling_factorial(n - 1, n - k)
def T(n, k): return (sum((-1)^(m + k) * binomial(n + k, n + m) * Lah(n + m, m)
        for m in range(k + 1)))
for n in range(8): print([T(n, k) for k in range(n+1)])

T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * L(n + m, m), where L denotes the unsigned Lah numbers A271703.
T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!.
T(n, k) = (2*(n + k - 1))*T(n-1, k-1) + (n + 2*k - 1)*T(n-1, k) with suitable boundary conditions (from Deb and Sokal). - Peter Luschny, Aug 06 2025

New name using a formula of Deb and Sokal by Peter Luschny, Aug 06 2025

A360205 Triangle read by rows. T(n, k) = (-1)^(n-k)(k+1)binomial(n, k)*pochhammer(1-n, n-k).

Original entry on oeis.org

1, 0, 2, 0, 4, 3, 0, 12, 18, 4, 0, 48, 108, 48, 5, 0, 240, 720, 480, 100, 6, 0, 1440, 5400, 4800, 1500, 180, 7, 0, 10080, 45360, 50400, 21000, 3780, 294, 8, 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9, 0, 725760, 4354560, 6773760, 4233600, 1270080, 197568, 16128, 648, 10

Offset: 0

Peter Luschny, Feb 08 2023

A refinement of the number of partial permutations of an n-set (A002720).

Also the coefficients of a shifted derivative of the unsigned Lah polynomials (A271703).

			Triangle T(n, k) starts:
[0] 1;
[1] 0,     2;
[2] 0,     4,      3;
[3] 0,    12,     18,      4;
[4] 0,    48,    108,     48,      5;
[5] 0,   240,    720,    480,    100,     6;
[6] 0,  1440,   5400,   4800,   1500,   180,    7;
[7] 0, 10080,  45360,  50400,  21000,  3780,  294,   8;
[8] 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9;

Cf. A052849 (column 1), A045991 (subdiagonal), A002720 (row sums), A271703.

Cf. A069138 (Stirling2 counterpart), A360174 (Stirling1 counterpart).

T := (n, k) -> (-1)^(n - k)*(k + 1)*binomial(n, k)*pochhammer(1 - n, n - k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

A371080 Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 28, 12, 1, 0, 280, 160, 24, 1, 0, 3640, 2520, 520, 40, 1, 0, 58240, 46480, 11880, 1280, 60, 1, 0, 1106560, 987840, 295960, 40040, 2660, 84, 1, 0, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1

Offset: 0

Peter Luschny, Mar 12 2024

			Triangle starts:
[0] 1;
[1] 0,       1;
[2] 0,       4,      1;
[3] 0,      28,     12,      1;
[4] 0,     280,    160,     24,     1;
[5] 0,    3640,   2520,    520,    40,    1;
[6] 0,   58240,  46480,  11880,  1280,   60,  1;
[7] 0, 1106560, 987840, 295960, 40040, 2660, 84, 1;

Peter Luschny, The Bell transform.

Variant: A035469.
Cf. A264428, A371076, A371077.
Cf. A007559, A144346.
Cf. A035342, A049029, A271703.

a := n -> mul(select(k -> k mod 3 = 1, [seq(1..3*(n + 1))])): BellMatrix(a, 9);
# Alternative:
BellMatrix(n -> coeff(series((1/x)*hypergeom([1, 1/3], [], 3*x),x, 22), x, n), 9);
# Recurrence:
T := proc(n, k) option remember; if k = n then 1 elif k = 0 then 0 else
T(n - 1, k - 1) + (3*(n - 1) + k) * T(n - 1, k) fi end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Mar 13 2024

T(n, k) = sum(j=k, n, 3^(n-j)*abs(stirling(n, j, 1))*stirling(j, k, 2)); \\ Seiichi Manyama, Apr 19 2025

T(n, k) = BellMatrix([x^n] hypergeom2F0([1, 1/3], [], 3*x) / x).
T(n, k) = A371076(n, k) / k!.
From Werner Schulte, Mar 13 2024: (Start)
T(n, k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1} (3*j + i)) / (k!).
T(n, k) = T(n-1, k-1) + (3*(n - 1) + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = 1 for n >= 0. (End)
From Seiichi Manyama, Apr 19 2025: (Start)
T(n,k) = Sum_{j=k..n} 3^(n-j) * |Stirling1(n,j)| * Stirling2(j,k).
E.g.f. of column k (with leading zeros): (1/(1 - 3*x)^(1/3) - 1)^k / k!. (End)

A373426 Triangle read by rows: Coefficients of the polynomials L(n, x) * EZ(n, x), where L denote the unsigned Lah polynomials and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 12, 7, 1, 0, 24, 108, 144, 73, 15, 1, 0, 120, 1080, 2640, 2660, 1221, 267, 27, 1, 0, 720, 11880, 48720, 82980, 67350, 28321, 6344, 751, 44, 1, 0, 5040, 146160, 955080, 2529240, 3262350, 2245782, 870283, 195074, 25267, 1831, 68, 1

Offset: 0

Peter Luschny, Jun 07 2024

			Tracing the computation:
  0: [1] *       [1] =                [1]
  1: [1] *       [0,  1] =            [0,  1]
  2: [1] *       [0,  2,  1] =        [0,  2,   1]
  3: [1, 1] *    [0,  6,  6,  1] =    [0,  6,  12,   7,  1]
  4: [1, 3, 1] * [0, 24, 36, 12, 1] = [0, 24, 108, 144, 73, 15, 1]

Peter Luschny, Illustrating the polynomials.

Cf. A271703 (Lah), A205497 (zig-zag Eulerian), A373425 (row sums).

# Using function EZP from A373432.
EZP((n, k) -> ifelse(n=k, 1, binomial(n-1, k-1)*n!/k!), 7);

A059374 Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.

Original entry on oeis.org

1, 3, 2, 13, 18, 6, 73, 156, 108, 24, 501, 1460, 1560, 720, 120, 4051, 15030, 21900, 15600, 5400, 720, 37633, 170142, 315630, 306600, 163800, 45360, 5040, 394353, 2107448, 4763976, 5891760, 4292400, 1834560, 423360, 40320

Offset: 1

Vladeta Jovovic, Jan 28 2001

L'(n, i) are unsigned Lah numbers (Cf. A008297).

			Triangle begins:
  [1],
  [3, 2],
  [13, 18, 6],
  [73, 156, 108, 24],
  [501, 1460, 1560, 720, 120],
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. T(n, 0) = A000262, A025168 (row sums), A000012 (alternating row sums), A059110.

Cf. A008297, A271703.

t[n_, k_] := Sum[ Binomial[n-1, n-i-1]*n!/(n-i)!*Binomial[i, k], {i, 0, n}]; Table[t[n, k], {n, 1, 8}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Mar 22 2013 *)

for(n=1,10, for(k=0,n-1, print1(sum(j=0,n, binomial(j,k)* binomial(n-1,n-j-1)*n!/(n-j)!), ", "))) \\ G. C. Greubel, Jan 29 2018

E.g.f.: exp(x/(1-(1+y)*x))/(1-(1+y)*x)^2. - Vladeta Jovovic, May 10 2003

cp's OEIS Frontend

A290596 Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].

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A134432 Sum of entries in all the arrangements of the set {1,2,...,n} (to n=0 there corresponds the empty set).

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A292219 Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,3].

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A290598 Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,2].

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A349776 Triangle read by rows: T(n,k) is the number of partitions of set [n] into a set of at most k lists, with 0 <= k <= n. Also called broken permutations.

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A357367 Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.

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A360205 Triangle read by rows. T(n, k) = (-1)^(n-k)*(k+1)*binomial(n, k)*pochhammer(1-n, n-k).

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A360205 Triangle read by rows. T(n, k) = (-1)^(n-k)(k+1)binomial(n, k)*pochhammer(1-n, n-k).