A203412 -id:A203412 - cp's OEIS frontend

A004747 Triangle read by rows: the Bell transform of the triple factorial numbers A008544 without column 0.

1, 2, 1, 10, 6, 1, 80, 52, 12, 1, 880, 600, 160, 20, 1, 12320, 8680, 2520, 380, 30, 1, 209440, 151200, 46480, 7840, 770, 42, 1, 4188800, 3082240, 987840, 179760, 20160, 1400, 56, 1, 96342400, 71998080, 23826880, 4583040, 562800, 45360, 2352, 72, 1

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A048966; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n,m) = S2p(-2; n,m), a member of a sequence of triangles including S2p(-1; n,m) = A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) = A008277(n, m) (Stirling 2nd kind). T(n,1)= A008544(n-1).
T(n,m), n>=m>=1, enumerates unordered n-vertex m-forests composed of m plane (aka ordered) increasing (rooted) trees where vertices of out-degree r>=0 come in r+1 different types (like an (r+1)-ary vertex). Proof from the e.g.f. of the first column Y(z) = 1 - (1-3*x)^(1/3) and the F. Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0) = 0, with out-degree o.g.f. phi(w)=1/(1-w)^2. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of the triple factorial numbers A008544 which adds a first column (1,0,0 ...) on the left side of the triangle. For the definition of the Bell transform see A264428. See A051141 for the triple factorial numbers A032031 and A203412 for the triple factorial numbers A007559 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 21 2015

			Triangle begins:
       1;
       2,      1;
      10,      6,     1;
      80,     52,    12,    1;
     880,    600,   160,   20,   1;
   12320,   8680,  2520,  380,  30,  1;
  209440, 151200, 46480, 7840, 770, 42, 1;
Tree combinatorics for T(3,2)=6: Consider first the unordered forest of m=2 plane trees with n=3 vertices, namely one vertex with out-degree r=0 (root) and two different trees with two vertices (one root with out-degree r=1 and a leaf with r=0). The 6 increasing labelings come then from the forest with rooted (x) trees x, o-x (1,(3,2)), (2,(3,1)) and (3,(2,1)) and similarly from the second forest x, x-o (1,(2,3)), (2,(1,3)) and (3,(1,2)).

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
F. 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.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) #09.3.3.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Index entries for sequences related to Bessel functions or polynomials

Cf. A015735 (row sums).
Cf. A007559, A008277, A008544, A032031, A039683.
Cf. A048966, A051141, A132062, A203412, A264428.
Triangles with the recurrence T(n,k) = (m*(n-1)-k)*T(n-1,k) + T(n-1,k-1): A010054 (m=1), A001497 (m=2), this sequence (m=3), A000369 (m=4), A011801 (m=5), A013988 (m=6).

function T(n,k) // T = A004747
  if k eq 0 then return 0;
  elif k eq n then return 1;
  else return (3*(n-1)-k)*T(n-1,k) + T(n-1,k-1);
  end if;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2023

T := (n, m) -> 3^n/m!*(1/3*m*GAMMA(n-1/3)*hypergeom([1-1/3*m, 2/3-1/3*m, 1/3-1/3*m], [2/3, 4/3-n], 1)/GAMMA(2/3)-1/6*m*(m-1)*GAMMA(n-2/3)*hypergeom( [1-1/3*m, 2/3-1/3*m, 4/3-1/3*m], [4/3, 5/3-n], 1)/Pi*3^(1/2)*GAMMA(2/3)):
for n from 1 to 6 do seq(simplify(T(n,k)),k=1..n) od;
# Karol A. Penson, Feb 06 2004
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(3*k+2, k=(0..n-1)), 9); # Peter Luschny, Jan 29 2016

(* First program *)
T[1,1]= 1; T[, 0]= 0; T[0, ]= 0; T[n_, m_]:= (3*(n-1)-m)*T[n-1, m]+T[n-1, m-1];
Flatten[Table[T[n, m], {n,12}, {m,n}] ][[1 ;; 45]] (* Jean-François Alcover, Jun 16 2011, after recurrence *)
(* Second program *)
f[n_, m_]:= m/n Sum[Binomial[k, n-m-k] 3^k (-1)^(n-m-k) Binomial[n+k-1, n-1], {k, 0, n-m}]; Table[n! f[n, m]/(m! 3^(n-m)), {n,12}, {m,n}]//Flatten (* Michael De Vlieger, Dec 23 2015 *)
(* Third program *)
rows = 12;
T[n_, m_]:= BellY[n, m, Table[Product[3k+2, {k, 0, j-1}], {j, 0, rows}]];
Table[T[n, m], {n,rows}, {m,n}]//Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses [bell_transform from A264428]
triplefactorial = lambda n: prod(3*k+2 for k in (0..n-1))
def A004747_row(n):
    trifact = [triplefactorial(k) for k in (0..n)]
    return bell_transform(n, trifact)
[A004747_row(n) for n in (0..10)] # Peter Luschny, Dec 21 2015

T(n, m) = n!*A048966(n, m)/(m!*3^(n-m));

T(n+1, m) = (3*n-m)*T(n, m)+ T(n, m-1), for n >= m >= 1, with T(n, m) = 0, for n

E.g.f. of m-th column: ( 1 - (1-3*x)^(1/3) )^m/m!.

Sum_{k=1..n} T(n, k) = A015735(n).

For a formula expressed as special values of hypergeometric functions 3F2 see the Maple program below. - Karol A. Penson, Feb 06 2004

T(n,1) = A008544(n-1). - Peter Luschny, Dec 23 2015

New name from Peter Luschny, Dec 21 2015

A051141 Triangle read by rows: a(n, m) = S1(n, m)*3^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).

Original entry on oeis.org

1, -3, 1, 18, -9, 1, -162, 99, -18, 1, 1944, -1350, 315, -30, 1, -29160, 22194, -6075, 765, -45, 1, 524880, -428652, 131544, -19845, 1575, -63, 1, -11022480, 9526572, -3191076, 548289, -52920, 2898, -84, 1, 264539520, -239660208

Offset: 1

Wolfdieter Lang

Previous name was: Generalized Stirling number triangle of first kind.
a(n,m) = R_n^m(a=0,b=3) in the notation of the given reference.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 3*j), n >= 1 and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
This is the signed Stirling1 triangle with diagonals d>=0 (main diagonal d=0) scaled with 3^d.
Exponential Riordan array [1/(1 + 3*x), log(1 + 3*x)/3]. The unsigned triangle is [1/(1 - 3*x), log(1/(1 - 3*x)^(1/3))]. - Paul Barry, Apr 29 2009
Also the Bell transform of the triple factorial numbers A032031 which adds a first column (1, 0, 0 ...) on the left side of the triangle and computes the unsigned values. For the definition of the Bell transform, see A264428. See A004747 for the triple factorial numbers A008544 and A203412 for the triple factorial numbers A007559 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 21 2015

			Triangle starts:
       1;
      -3,       1;
      18,      -9,      1;
    -162,      99,    -18,      1;
    1944,   -1350,    315,    -30,    1;
  -29160,   22194,  -6075,    765,  -45,   1;
  524880, -428652, 131544, -19845, 1575, -63, 1;
---
Row polynomial E(3,x) = 18*x-9*x^2+x^3.
From _Paul Barry_, Apr 29 2009: (Start)
The unsigned array [1/(1 - 3*x), log(1/(1 - 3*x)^(1/3))] has production matrix
    3,    1;
    9,    6,    1;
   27,   27,    9,   1;
   81,  108,   54,  12,   1;
  243,  405,  270,  90,  15,  1;
  729, 1458, 1215, 540, 135, 18, 1;
  ...
which is A007318^{3} beheaded (by viewing A007318 as a lower triangular matrix). See the comment above. (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Wolfdieter Lang, First 10 rows.
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. 77 (1962), 1-77.

First (m=1) column sequence is: A032031(n-1).
Row sums (signed triangle): A008544(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A007559(n).
Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051142 (b=4).
Cf. A039683, A132062, A264428.

a[n_, m_] /; n >= m >= 1 := a[n, m] = a[n-1, m-1] - 3(n-1)*a[n-1, m]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]] (* _Jean-François Alcover, Jun 01 2011, after formula *)
Table[StirlingS1[n, m]*3^(n - m), {n, 1, 10}, {m, 1, n}]//Flatten (* G. C. Greubel, Oct 24 2017 *)

for(n=1,10, for(m=1,n, print1(stirling(n,m,1)*3^(n-m), ", "))) \\ G. C. Greubel, Oct 24 2017

# uses[bell_transform from A264428]
triplefactorial = lambda n: 3^n*factorial(n)
def A051141_row(n):
    trifact = [triplefactorial(k) for k in (0..n)]
    return bell_transform(n, trifact)
[A051141_row(n) for n in (0..8)] # Peter Luschny, Dec 21 2015

a(n, m) = a(n-1, m-1) - 3*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) = 0 for n < m; a(n, 0) = 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 3*x)/3)^m/m!.
|a(n,1)| = A032031(n-1). - Peter Luschny, Dec 23 2015

Name clarified using a formula of the author by Peter Luschny, Dec 23 2015

A265606 Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 45, 23, 6, 1, 0, 585, 275, 65, 10, 1, 0, 9945, 4435, 990, 145, 15, 1, 0, 208845, 89775, 19285, 2730, 280, 21, 1, 0, 5221125, 2183895, 456190, 62965, 6370, 490, 28, 1, 0, 151412625, 62002395, 12676265, 1715490, 171255, 13230, 798, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[1],
[0, 1],
[0, 1, 1],
[0, 5, 3, 1],
[0, 45, 23, 6, 1],
[0, 585, 275, 65, 10, 1],
[0, 9945, 4435, 990, 145, 15, 1],
[0, 208845, 89775, 19285, 2730, 280, 21, 1],

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Peter Luschny, The Bell transform

Cf. A007696, A264428, A264429.

Bell transforms of other multifactorials are: A000369, A004747, A039683, A051141, A051142, A119274, A132062, A132393, A203412.

(* The function BellMatrix is defined in A264428. *)
rows = 10;
M = BellMatrix[Pochhammer[1/4, #] 4^# &, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 23 2019 *)

# uses[bell_transform from A264428]
def A265606_row(n):
    multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
    mfact = [multifact_4_1(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A265606_row(n) for n in (0..7)]

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A004747 Triangle read by rows: the Bell transform of the triple factorial numbers A008544 without column 0.

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A051141 Triangle read by rows: a(n, m) = S1(n, m)*3^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).

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A265606 Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).

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Mathematica

Sage