A132062 -id:A132062 - cp's OEIS frontend

A039683 Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).

1, -2, 1, 8, -6, 1, -48, 44, -12, 1, 384, -400, 140, -20, 1, -3840, 4384, -1800, 340, -30, 1, 46080, -56448, 25984, -5880, 700, -42, 1, -645120, 836352, -420224, 108304, -15680, 1288, -56, 1, 10321920, -14026752, 7559936, -2153088, 359184, -36288, 2184, -72, 1

Offset: 1

Wouter Meeussen

T(n,m) = R_n^m(a=0,b=2) in the notation of the given reference.
Exponential Riordan array [1/(1+2x),log(1+2x)/2]. The unsigned triangle is [1/(1-2x),log(1/sqrt(1-2x))]. - Paul Barry_, Apr 29 2009
The n-th row is related to the expansion of z^(-2n)*(z^3 d/dz)^n in polynomials of the Euler operator D=(z d/dz). E.g., z^(-6)(z^3 d/dz)^3 = D^3 + 6 D^2 + 8 D. See Copeland link for relations to Bell / Exponential / Touchard polynomial operators. - Tom Copeland, Nov 14 2013
A refinement of this array is given by A231846. - Tom Copeland, Nov 15 2013
Also the Bell transform of the double factorial of even numbers A000165 except that the values are unsigned and in addition a first column (1,0,0 ...) is added on the left side of the triangle. For the Bell transform of the double factorial of odd numbers A001147 see A132062. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015
The signed triangle is also the inverse Bell transform of A000079 (see Luschny link). - John Keith, Nov 24 2020

			Triangle starts:
  {1},
  {2,1},
  {8,6,1},
  {48,44,12,1},
  ...
From _Paul Barry_, Apr 29 2009: (Start)
The unsigned triangle [1/(1-2x),log(1/sqrt(1-2x))] has production matrix:
  2, 1,
  4, 4, 1,
  8, 12, 6, 1,
  16, 32, 24, 8, 1,
  32, 80, 80, 40, 10, 1,
  64, 192, 240, 160, 60, 12, 1
which is A007318^{2} beheaded. (End)

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, Addendum to Mathemagical Forests.
P. Feijão, F. V. Martinez, and A. Thévenin, On the distribution of cycles and paths in multichromosomal breakpoint graphs and the expected value of rearrangement distance, BMC Bioinformatics 16:Suppl19 (2015), S1. doi:10.1186/1471-2105-16-S19-S1
Lisa Glaser, Causal set actions in various dimensions, J. Phys.: Conf. Ser. 306 (2011), 012041.
Wolfdieter Lang, First 9 rows and comment.
Peter Luschny, The Bell transform
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres relies aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

First column (unsigned triangle) is (2(n-1))!! = 1, 2, 8, 48, 384...= A000165(n-1) and the row sums (unsigned) are (2n-1)!! = 1, 3, 15, 105, 945... = A001147(n-1).
Cf. A051141, A051142.
Cf. A000165, A132062, A264428.
Cf. A038207.

Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 6} ]

# uses[bell_transform from A264428]
# Unsigned values and an additional first column (1,0,0,...).
def A039683_unsigned_row(n):
    a = sloane.A000165
    dblfact = a.list(n)
    return bell_transform(n, dblfact)
[A039683_unsigned_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015

T(n, m) = T(n-1, m-1) - 2*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) := 0, n

E.g.f. for m-th column of signed triangle: (((log(1+2*x))/2)^m)/m!.

E.g.f.: (1+2*x)^(y/2). O.g.f. for n-th row of signed triangle: Sum_{m=0..n} Stirling1(n, m)*2^(n-m)*x^m. - Vladeta Jovovic, Feb 11 2003

T(n, m) = S1(n, m)*2^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

The production matrix below is A038207 with the first row removed. With the initial index n = 0, the associated differential raising operator is R = e^(2D)*x = (2+x)*e^(2D) with D = d/dx, i.e., R p_n(x) = p_(n+1)(x) where p_n(x) is the n-th unsigned row polynomial and p_0(x) = 1, so p_(n+1)(x) = (2+x) * p_n(2+x). - Tom Copeland, Oct 11 2016

Additional comments from Wolfdieter Lang

Title revised by Tom Copeland, Dec 21 2013

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

Original entry on oeis.org

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

A203412 Triangle read by rows, a(n,k), n>=k>=1, which represent the s=3, h=1 case of a two-parameter generalization of Stirling numbers arising in conjunction with normal ordering.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 28, 19, 6, 1, 280, 180, 55, 10, 1, 3640, 2260, 675, 125, 15, 1, 58240, 35280, 10360, 1925, 245, 21, 1, 1106560, 658000, 190680, 35385, 4620, 434, 28, 1, 24344320, 14266560, 4090240, 756840, 100065, 9828, 714, 36, 1

Offset: 1

Mark Shattuck, Jan 01 2012

Also the Bell transform of the triple factorial numbers A007559 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 A004747 for the triple factorial numbers A008544 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 23 2015

			Triangle starts:
[    1]
[    1,     1]
[    4,     3,     1]
[   28,    19,     6,    1]
[  280,   180,    55,   10,   1]
[ 3640,  2260,   675,  125,  15,  1]
[58240, 35280, 10360, 1925, 245, 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.
T. Mansour, M. Schork, and M. Shattuck, On a new family of generalized Stirling and Bell numbers, Electron. J. Combin. 18 (2011) #P77 (33 pp.).
Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771.

Cf. A007559, A032031, A039683, A051141, A132062, A264428.

A203412 := (n,k) -> (n!*3^n)/(k!*2^k)*add((-1)^j*binomial(k,j)*binomial(n-2*j/3-1, n), j=0..k): seq(seq(A203412(n,k),k=1..n),n=1..9); # Peter Luschny, Dec 21 2015

Table[(n! 3^n)/(k! 2^k) Sum[ (-1)^j Binomial[k, j] Binomial[n - 2 j/3 - 1, n], {j, 0, k}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Dec 23 2015 *)

# uses[bell_transform from A264428]
triplefactorial = lambda n: prod(3*k + 1 for k in (0..n-1))
def A203412_row(n):
    trifact = [triplefactorial(k) for k in (0..n)]
    return bell_transform(n, trifact)
[A203412_row(n) for n in (0..8)] # Peter Luschny, Dec 21 2015

(1) Is given by the recurrence relation
  a(n+1,k) = a(n,k-1)+(3*n-2*k)*a(n,k) if n>=0 and k>=1, along with the initial values a(n,0) = delta_{n,0} and a(0,k) = delta_{0,k} for all n,k>=0.
(2) Is given explicitly by
  a(n,k) = (n!*3^n)/(k!*2^k)*Sum{j=0..k} (-1)^j*C(k,j)*C(n-2*j/3-1,n) for all n>=k>=1.
a(n,1) = A007559(n-1). - Peter Luschny, Dec 21 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)]

A122850 Exponential Riordan array (1, sqrt(1+2x)-1).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 3, -3, 1, 0, -15, 15, -6, 1, 0, 105, -105, 45, -10, 1, 0, -945, 945, -420, 105, -15, 1, 0, 10395, -10395, 4725, -1260, 210, -21, 1, 0, -135135, 135135, -62370, 17325, -3150, 378, -28, 1, 0, 2027025, -2027025, 945945, -270270, 51975, -6930, 630, -36, 1

Offset: 0

Paul Barry, Sep 14 2006

Inverse of number triangle A122848. Entries are Bessel polynomial coefficients. Row sums are A000806.

Also the inverse Bell transform of the sequence "g(n) = 1 if n<2 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins
  1
  0 1
  0 -1 1
  0 3 -3 1
  0 -15 15 -6 1
  0 105 -105 45 -10 1
  0 -945 945 -420 105 -15 1
  0 10395 -10395 4725 -1260 210 -21 1
  0 -135135 135135 -62370 17325 -3150 378 -28 1
  0 2027025 -2027025 945945 -270270 51975 -6930 630 -36 1
  0 -34459425 34459425 -16216200 4729725 -945945 135135 -13860 990 -45 1
  ...

P. Bala, The white diamond product of power series
Orli Herscovici, Study of the p,q-deformed Touchard polynomials, arXiv:1904.07674 [math.CO], 2019.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wikipedia, Bessel polynomials
S. Willerton, The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials, arXiv:1708.03227v1 [math.MG], 2017.

Cf. A000806, A001497, A008277, A039757, A122848, A132062, A144301.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-1)^n*doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[Function[n, (-1)^n (2n-1)!!], rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<2 else 0, 12).inverse() # Peter Luschny, Jan 19 2016

T(n,k) = (-1)^(n-k)*A132062(n,k). - Philippe Deléham, Nov 06 2011
Triangle equals the matrix product A039757*A008277. Dobinski-type formula for the row polynomials: R(n,x) = x*exp(-x)*Sum_{k = 0..inf} (k-1)*(k-3)*(k-5)*...*(k-(2*n-3))*x^k/k! for n >= 1. Cf. A001497. - Peter Bala, Jun 23 2014
From Peter Bala, Jan 09 2018: (Start)
Alternative Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-(2*n-2))*x^k/k!.
Equivalently, R(n,x) = x o (x-2) o (x-4) o...o (x-(2*n-2)), where o denotes the white diamond product of polynomials. See the Bala link for the definition and details.
The white diamond products (x-1) o (x-3) o...o (x-(2*n-3)) give the row polynomials of the array with a factor of x removed.
If d is the first derivative operator f -> d/dx(f(x)) and D is the operator f(x) -> 1/x*d/dx(f(x)) then x^(2*n)*D^n = R(n,x*d), with the understanding that (x*d)^k is to interpreted as the operator f(x) -> x^k*d^k(f(x))/dx^k. (End)
Sum_{k=0..n} (-1)^(n+k) * T(n,k) = A144301(n). - Alois P. Heinz, Aug 31 2022

More terms from Alois P. Heinz, Aug 31 2022

A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).

Original entry on oeis.org

1, 1, -2, 3, -4, 2, 15, -18, 9, -2, 105, -120, 60, -16, 2, 945, -1050, 525, -150, 25, -2, 10395, -11340, 5670, -1680, 315, -36, 2, 135135, -145530, 72765, -22050, 4410, -588, 49, -2, 2027025, -2162160, 1081080, -332640, 69300, -10080, 1008, -64, 2

Offset: 0

Johannes W. Meijer, May 31 2018

The function f(u, p) = (1/2)*(1+1/(sqrt(1-u^2))) * exp(p*sqrt(1-u^2)) was found while studying the Fresnel-Kirchhoff and the Rayleigh-Sommerfeld theories of diffraction, see the Meijer link.
The Taylor expansion of f(u, p) leads to the number triangle T(n, k), see the example section.
Normalization of the triangle terms, dividing the T(n, k) by T(n-k, 0), leads to A084534.
The row sums equal A003436, n >= 2, respectively A231622, n >= 1.

			The first few terms of the Taylor expansion of f(u; p) are:
f(u, p) = exp(p) * (1 + (1-2*p) * u^2/4 + (3-4*p+2*p^2) * u^4/16 + (15-18*p+9*p^2-2*p^3) * u^6/96 + (105-120*p+60*p^2-16*p^3+2*p^4) * u^8/768 + ... )
The first few rows of the T(n, k) triangle are:
n=0:     1
n=1:     1,     -2
n=2:     3,     -4,    2
n=3:    15,    -18,    9,    -2
n=4:   105,   -120,   60,   -16,   2
n=5:   945,  -1050,  525,  -150,  25,  -2
n=6: 10395, -11340, 5670, -1680, 315, -36, 2

J. W. Goodman, Introduction to Fourier Optics, 1996.
A. Papoulis, Systems and Transforms with Applications in Optics, 1968.

Andrew Howroyd, Rows n=0..50 of triangle, flattened
M. J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Optics Communications, Vol. 25, nr. 1, pp. 26-30, 1978.
H. J. Butterweck, General theory of linear, coherent optical data processing systems, Journal of the Optical Society of America, Vol. 67, nr. 1, pp. 60-70, 1977.
J. W. Meijer, A note on optical diffraction, 1979.

Cf. A003436, A231622, A032184, A084534, A127674, A132062, A001497.
Cf. Related to the left hand columns: A001147, A001193, A261065.
Cf. Related to the right hand columns: A280560, A162395, A006011, A040977, A053347, A054334, A266561.

[[n le 0 select 1 else (-1)^k*2^(k-n+1)*Factorial(2*n-k-1)*Binomial(n, k)/Factorial(n-1): k in [0..n]]: n in [1..10]]; // G. C. Greubel, Nov 08 2018

T := proc(n, k): if n=0 then 1 else (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!) fi: end: seq(seq(T(n, k), k=0..n), n=0..8);

Table[If[n==0 && k==0,1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!)], {n, 0, 10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 08 2018 *)

T(n,k) = {if(n==0, 1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 08 2018

T(n, k) = (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*A001147(n-k)*A084534(n, k), n >= 0 and 0 <= k <= n.
T(n, k) = 2^(2*(k-n)+1)*A001147(n-k)*A127674(n, n-k), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*(A001497(n, k) + A132062(n, k)), n >= 1, T(0,0) = 1.

A369746 Expansion of e.g.f. exp( 3 * (1-sqrt(1-2*x)) ).

Original entry on oeis.org

1, 3, 12, 63, 423, 3528, 35559, 422901, 5817744, 91072269, 1600588269, 31230827532, 670252672593, 15696888917427, 398454496989012, 10899543418960167, 319672849622745951, 10007954229075765984, 333139545206104991031, 11749955670275356579941

Offset: 0

Seiichi Manyama, Jan 30 2024

nonn

Cf. A107104, A144301, A132062.

Cf. A369722.

# The row polynomials of A132062 evaluated at x = 3.
T := proc(n, k) option remember; if k = 0 then 0^n elif n < k then 0
else (2*(n - 1) - k)*T(n - 1, k) + T(n - 1, k - 1) fi end:
seq(add(T(n, k)*3^k, k = 0..n), n = 0..19);  # Peter Luschny, Apr 25 2024

With[{nn=20},CoefficientList[Series[Exp[3(1-Sqrt[1-2x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 14 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3*(1-sqrt(1-2*x)))))

a(0) = 1; a(n) = Sum_{k=0..n-1} 3^(n-k) * (n-1+k)! / (2^k * k! * (n-1-k)!).

a(n) = (2*n-3)*a(n-1) + 9*a(n-2).

Showing 1-9 of 9 results.

cp's OEIS Frontend

A376872 a(n) = n! * 2^(-n) * binomial(3*n - 1, 2*n) * binomial(2*n, n). Central terms of the Bessel triangle A132062.

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A039683 Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).

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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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A203412 Triangle read by rows, a(n,k), n>=k>=1, which represent the s=3, h=1 case of a two-parameter generalization of Stirling numbers arising in conjunction with normal ordering.

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

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A122850 Exponential Riordan array (1, sqrt(1+2x)-1).

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A376872 a(n) = n! * 2^(-n) * binomial(3n - 1, 2n) * binomial(2*n, n). Central terms of the Bessel triangle A132062.

A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).