A264429 -id:A264429 - cp's OEIS frontend

A111596 The matrix inverse of the unsigned Lah numbers A271703.

1, 0, 1, 0, -2, 1, 0, 6, -6, 1, 0, -24, 36, -12, 1, 0, 120, -240, 120, -20, 1, 0, -720, 1800, -1200, 300, -30, 1, 0, 5040, -15120, 12600, -4200, 630, -42, 1, 0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1, 0, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016, -72, 1

Offset: 0

Wolfdieter Lang, Aug 23 2005

Also the associated Sheffer triangle to Sheffer triangle A111595.
Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-1,x), which equals (-1)^n * Lag(n,x,-1) below. Lag(n,Lag(.,x,-1),-1) = x^n evaluated umbrally, i.e., with (Lag(.,x,-1))^k = Lag(k,x,-1). - Tom Copeland, Apr 26 2014
Without row n=0 and column m=0 this is, up to signs, the Lah triangle A008297.
The unsigned column sequences are (with leading zeros): A000142, A001286, A001754, A001755, A001777, A001778, A111597-A111600 for m=1..10.
The row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m, together with the row polynomials s(n,x) of A111595 satisfy the exponential (or binomial) convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), n>=0.
Exponential Riordan array [1,x/(1+x)]. Inverse of the exponential Riordan array [1,x/(1-x)], which is the unsigned version of A111596. - Paul Barry, Apr 12 2007
For the unsigned subtriangle without column number m=0 and row number n=0, see A105278.
Unsigned triangle also matrix product |S1|*S2 of Stirling number matrices.
The unsigned row polynomials are Lag(n,-x,-1), the associated Laguerre polynomials of order -1 with negated argument. See Gradshteyn and Ryzhik, Abramowitz and Stegun and Rota (Finite Operator Calculus) for extensive formulas. - Tom Copeland, Nov 17 2007, Sep 09 2008
An infinitesimal matrix generator for unsigned A111596 is given by A132792. - Tom Copeland, Nov 22 2007
From the formalism of A132792 and A133314 for n > k, unsigned A111596(n,k) = a(k) * a(k+1)...a(n-1) / (n-k)! = a generalized factorial, where a(n) = A002378(n) = n-th term of first subdiagonal of unsigned A111596. Hence Deutsch's remark in A002378 provides an interpretation of A111596(n,k) in terms of combinations of certain circular binary words. - Tom Copeland, Nov 22 2007
Given T(n,k)= A111596(n,k) and matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 27 2008
Operationally, the unsigned row polynomials may be expressed as p_n(:xD:) = x*:Dx:^n*x^{-1}=x*D^nx^n*x^{-1}= n!*binomial(xD+n-1,n) = (-1)^n n! binomial(-xD,n) = n!L(n,-1,-:xD:), where, by definition, :AB:^n = A^nB^n for any two operators A and B, D = d/dx, and L(n,-1,x) is the Laguerre polynomial of order -1. A similarity transformation of the operators :Dx:^n generates the higher order Laguerre polynomials, which can also be expressed in terms of rising or falling factorials or Kummer's confluent hypergeometric functions (cf. the Mathoverflow post). -  Tom Copeland, Sep 21 2019

			Binomial convolution of row polynomials: p(3,x) = 6*x-6*x^2+x^3; p(2,x) = -2*x+x^2, p(1,x) = x, p(0,x) = 1,
together with those from A111595: s(3,x) = 9*x-6*x^2+x^3; s(2,x) = 1-2*x+x^2, s(1,x) = x, s(0,x) = 1; therefore
9*(x+y)-6*(x+y)^2+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = (6*y-6*y^2+y^3) + 3*x*(-2*y+y^2) + 3*(1-2*x+x^2)*y + 9*x-6*x^2+x^3.
From _Wolfdieter Lang_, Apr 28 2014: (Start)
The triangle a(n,m) begins:
n\m  0     1       2       3      4     5   6  7
0:   1
1:   0     1
2:   0    -2       1
3:   0     6      -6       1
4:   0   -24      36     -12      1
5:   0   120    -240     120    -20     1
6:   0  -720    1800   -1200    300   -30   1
7:   0  5040  -15120   12600  -4200   630 -42  1
...
For more rows see the link.
(End)

G. C. Greubel, Rows n=0..100 of triangle, flattened
Wolfdieter Lang, The first 11 rows of the triangle.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 4.
Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 6.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 20.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011) 11.6.7.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers; Generators, Inversion, and Matrix, Binomial, and Integral Transforms; Lagrange a la Lah
A. Hennessy and P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Mathoverflow, Pochhammer symbol of a differential, and hypergeometric polynomials, a question posed by Emilio Pisanty and answered by Tom Copeland, 2012.
J. Taylor, Counting words with Laguerre polynomials, DMTCS Proc., Vol. AS, 2013, p. 1131-1142. [_Tom Copeland_, Jan 08 2016] [Broken link]
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96. [_Tom Copeland_, Dec 20 2018]
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Row sums: A111884. Unsigned row sums: A000262.
A002868 gives maximal element (in magnitude) in each row.
Cf. A130561 for a natural refinement.
Cf. A264428, A264429, A271703 (unsigned).
Cf. A008297, A089231, A105278 (variants).
Cf. A002378, A008277, A132792, A133314, A132792, A001263.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::odd, -(n+1)!, (n+1)!), 9); # Peter Luschny, Jan 27 2016

a[0, 0] = 1; a[n_, m_] := ((-1)^(n-m))*(n!/m!)*Binomial[n-1, m-1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)
T[ n_, k_] := (-1)^n n! Coefficient[ LaguerreL[ n, -1, x], x, k]; (* Michael Somos, Dec 15 2014 *)
rows = 9;
t = Table[(-1)^(n+1) n!, {n, 1, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}]  // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

{T(n, k) = if( n<1 || k<1, n==0 && k==0, (-1)^n * n! * polcoeff( sum(k=1, n, binomial( n-1, k-1) * (-x)^k / k!), k))}; /* Michael Somos, Dec 15 2014 */

lah_number = lambda n, k: factorial(n-k)*binomial(n,n-k)*binomial(n-1,n-k)
A111596_row = lambda n: [(-1)^(n-k)*lah_number(n, k) for k in (0..n)]
for n in range(10): print(A111596_row(n)) # Peter Luschny, Oct 05 2014

# uses[inverse_bell_transform from A264429]
def A111596_matrix(dim):
    fact = [factorial(n) for n in (1..dim)]
    return inverse_bell_transform(dim, fact)
A111596_matrix(10) # Peter Luschny, Dec 20 2015

E.g.f. m-th column: ((x/(1+x))^m)/m!, m>=0.
E.g.f. for row polynomials p(n, x) is exp(x*y/(1+y)).
a(n, m) = ((-1)^(n-m))*|A008297(n, m)| = ((-1)^(n-m))*(n!/m!)*binomial(n-1, m-1), n>=m>=1; a(0, 0)=1; else 0.
a(n, m) = -(n-1+m)*a(n-1, m) + a(n-1, m-1), n>=m>=0, a(n, -1):=0, a(0, 0)=1; a(n, m)=0 if n|a(n,m)| = Sum_{k=m..n} |S1(n,k)|*S2(k,m), n>=0. S2(n,m):=A048993. S1(n,m):=A048994. - Wolfdieter Lang, May 04 2007
From Tom Copeland, Nov 21 2011: (Start)
For this Lah triangle, the n-th row polynomial is given umbrally by
  (-1)^n n! binomial(-Bell.(-x),n), where Bell_n(-x)= exp(x)(xd/dx)^n exp(-x), the n-th Bell / Touchard / exponential polynomial with neg. arg., (cf. A008277). E.g., 2! binomial(-Bell.(-x),2) = -Bell.(-x)*(-Bell.(-x)-1) = Bell_2(-x)+Bell_1(-x) = -2x+x^2.
A Dobinski relation is (-1)^n n! binomial(-Bell.(-x),n)= (-1)^n n! e^x Sum_{j>=0} (-1)^j binomial(-j,n)x^j/j!= n! e^x Sum_{j>=0} (-1)^j binomial(j-1+n,n)x^j/j!. See the Copeland link for the relation to inverse Mellin transform. (End)
The n-th row polynomial is (-1/x)^n e^x (x^2*D_x)^n e^(-x). - Tom Copeland, Oct 29 2012
Let f(.,x)^n = f(n,x) = x!/(x-n)!, the falling factorial,and r(.,x)^n = r(n,x) = (x-1+n)!/(x-1)!, the rising factorial, then the Lah polynomials, Lah(n,t)= n!*Sum{k=1..n} binomial(n-1,k-1)(-t)^k/k! (extra sign factor on odd rows), give the transform Lah(n,-f(.,x))= r(n,x), and Lah(n,r(.,x))= (-1)^n * f(n,x). - Tom Copeland, Oct 04 2014
|T(n,k)| = Sum_{j=0..2*(n-k)} A254881(n-k,j)*k^j/(n-k)!. Note that A254883 is constructed analogously from A254882. - Peter Luschny, Feb 10 2015
The T(n,k) are the inverse Bell transform of [1!,2!,3!,...] and |T(n,k)| are the Bell transform of [1!,2!,3!,...]. See A264428 for the definition of the Bell transform and A264429 for the definition of the inverse Bell transform. - Peter Luschny, Dec 20 2015
Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates a shifted, signed Narayana matrix A001263. - Tom Copeland, Sep 23 2020

New name using a comment from Wolfdieter Lang by Peter Luschny, May 10 2021

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

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Apr 18 2008

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

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

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

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

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

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

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

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

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

Better name by Peter Bala, Apr 08 2013

Edited by Joerg Arndt, Apr 08 2013

A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -2, 3, 1, 0, 10, -5, 6, 1, 0, -80, 30, -5, 10, 1, 0, 880, -290, 45, 5, 15, 1, 0, -12320, 3780, -560, 35, 35, 21, 1, 0, 209440, -61460, 8820, -735, 0, 98, 28, 1, 0, -4188800, 1192800, -167300, 14700, -735, 0, 210, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[ 1]
[ 0,      1]
[ 0,      1,      1]
[ 0,     -2,      3,      1]
[ 0,     10,     -5,      6,      1]
[ 0,    -80,     30,     -5,     10,      1]
[ 0,    880,   -290,     45,      5,     15,      1]

Peter Luschny, The Bell transform
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.

Cf. A007696, A264428, A264429.

Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265605.

# uses[bell_transform from A264428]
def inverse_bell_matrix(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
    return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
print(inverse_bell_matrix(multifact_4_1, 8))

A265605 Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -1, 3, 1, 0, 3, -1, 6, 1, 0, -15, 5, 5, 10, 1, 0, 105, -35, 0, 25, 15, 1, 0, -945, 315, -35, 0, 70, 21, 1, 0, 10395, -3465, 490, -35, 70, 154, 28, 1, 0, -135135, 45045, -6895, 630, -105, 378, 294, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[ 1]
[ 0,    1]
[ 0,    1,    1]
[ 0,   -1,    3,    1]
[ 0,    3,   -1,    6,    1]
[ 0,  -15,    5,    5,   10,    1]
[ 0,  105,  -35,    0,   25,   15,    1]
[ 0, -945,  315,  -35,    0,   70,   21,    1]

Peter Luschny, The Bell transform
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.

Cf. A007559, A264428, A264429.

Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265604.

# uses[bell_transform from A264428]
def inverse_bell_matrix(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
    return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
multifact_3_1 = lambda n: prod(3*k + 1 for k in (0..n-1))
print(inverse_bell_matrix(multifact_3_1, 8))

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)]

A264431 Triangle read by rows, inverse Bell transform of second order Bell numbers (A187761).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, -3, 1, 0, -1, 7, -6, 1, 0, 2, -15, 25, -10, 1, 0, -8, 37, -90, 65, -15, 1, 0, 27, -133, 322, -350, 140, -21, 1, 0, -70, 587, -1330, 1757, -1050, 266, -28, 1, 0, 265, -2526, 6607, -9114, 7077, -2646, 462, -36, 1

Offset: 0

Peter Luschny, Nov 13 2015

			[ 1 ]
[ 0,     1 ]
[ 0,    -1,     1 ]
[ 0,     1,    -3,     1 ]
[ 0,    -1,     7,    -6,     1 ]
[ 0,     2,   -15,    25,   -10,     1 ]
[ 0,    -8,    37,   -90,    65,   -15,     1 ]
[ 0,    27,  -133,   322,  -350,   140,   -21,     1 ]
[ 0,   -70,   587, -1330,  1757, -1050,   266,   -28,     1 ]
[ 0,   265, -2526,  6607, -9114,  7077, -2646,   462,   -36,   1 ]

Peter Luschny, The Bell transform

Cf. A000110, A187761, A264428, A264429.

# uses[bell_transform from A264428, inverse_bell_transform from A264429]
def A264431_matrix(dim):
    uno = [1]*dim
    bell_numbers = [sum(bell_transform(n, uno)) for n in range(dim)]
    bell_number_2 = [sum(bell_transform(n, bell_numbers)) for n in range(dim)]
    return inverse_bell_transform(dim, bell_number_2)
A264431_matrix(10)

A264436 Triangle read by rows, inverse Bell transform of the complementary Bell numbers (A000587); T(n,k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 14, 15, 6, 1, 0, 89, 100, 45, 10, 1, 0, 716, 834, 405, 105, 15, 1, 0, 6967, 8351, 4284, 1225, 210, 21, 1, 0, 79524, 97596, 52220, 16009, 3080, 378, 28, 1, 0, 1041541, 1303956, 721674, 233268, 48699, 6804, 630, 36, 1

Offset: 0

Peter Luschny, Dec 01 2015

			Triangle starts:
1,
0,     1,
0,     1,     1,
0,     3,     3,     1,
0,    14,    15,     6,     1,
0,    89,   100,    45,    10,    1,
0,   716,   834,   405,   105,   15,   1,
0,  6967,  8351,  4284,  1225,  210,  21,  1,
0, 79524, 97596, 52220, 16009, 3080, 378, 28, 1

Peter Luschny, The Bell transform

Cf. A000587, A007549, A029768, A264428, A264429.

# uses[bell_transform from A264428, inverse_bell_transform from A264429]
def A264436_matrix(dim):
    uno = [1]*dim
    complementary_bell_numbers = [sum((-1)^n*b for (n, b) in enumerate (bell_transform(n, uno))) for n in (0..dim)]
    return inverse_bell_transform(dim, complementary_bell_numbers)
A264436_matrix(9)

Row sums are A029768(n-1) for n>=1.

T(n,1) = A007549(n) for n>=1.

A264434 Triangle read by rows, inverse Bell transform of the third-order Bell numbers, T(n,k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, -3, 1, 0, -1, 7, -6, 1, 0, 1, -15, 25, -10, 1, 0, 0, 31, -90, 65, -15, 1, 0, -7, -56, 301, -350, 140, -21, 1, 0, 33, 35, -938, 1701, -1050, 266, -28, 1, 0, -102, 423, 2485, -7686, 6951, -2646, 462, -36, 1, 0, 240, -3219, -3450, 31885

Offset: 0

Peter Luschny, Dec 02 2015

			[ 1]
[ 0,     1]
[ 0,    -1,     1]
[ 0,     1,    -3,     1]
[ 0,    -1,     7,    -6,      1]
[ 0,     1,   -15,    25,    -10,     1]
[ 0,     0,    31,   -90,     65,    -15,    1]
[ 0,    -7,   -56,   301,   -350,    140,   -21,  1]
[ 0,    33,    35,  -938,   1701,  -1050,   266, -28,  1]
[ 0,  -102,   423,  2485,  -7686,   6951, -2646, 462, -36, 1]

Cf. A048993, A264428, A264429, A264431.

# uses[bell_transform from A264428, inverse_bell_transform from A264429]
def A264434_matrix(dim):
    uno = [1]*dim
    bell_numbers = [sum(bell_transform(n, uno)) for n in range(dim)]
    bell_number_2 = [sum(bell_transform(n, bell_numbers)) for n in range(dim)]
    bell_number_3 = [sum(bell_transform(n, bell_number_2)) for n in range(dim)]
    return inverse_bell_transform(dim, bell_number_3)
A264434_matrix(10)

Showing 1-8 of 8 results.

cp's OEIS Frontend

A111596 The matrix inverse of the unsigned Lah numbers A271703.

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

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A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

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A265605 Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).

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

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A264431 Triangle read by rows, inverse Bell transform of second order Bell numbers (A187761).

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A264436 Triangle read by rows, inverse Bell transform of the complementary Bell numbers (A000587); T(n,k) for n>=0 and 0<=k<=n.

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A264434 Triangle read by rows, inverse Bell transform of the third-order Bell numbers, T(n,k) for n >= 0 and 0 <= k <= n.