A059340 -id:A059340 - cp's OEIS frontend

A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 15, 37, 31, 10, 1, 52, 151, 160, 75, 15, 1, 203, 674, 856, 520, 155, 21, 1, 877, 3263, 4802, 3556, 1400, 287, 28, 1, 4140, 17007, 28337, 24626, 11991, 3290, 490, 36, 1, 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786, 45, 1

Offset: 0

N. J. A. Sloane

Triangle a(n,k) read by rows; given by [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deléham's operator defined in A084938.
Exponential Riordan array [exp(exp(x)-1), exp(x)-1]. - Paul Barry, Jan 12 2009
Equal to A048993*A007318. - Philippe Deléham, Oct 31 2011
This lower unitriangular array is the L factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20U%20factor%20is%20A059098%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The U factor is A059098 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

			Triangle begins:
   1;
   1,  1;
   2,  3,  1;
   5, 10,  6,  1;
  15, 37, 31, 10,  1;
  ...
From _Paul Barry_, Jan 12 2009: (Start)
Production array begins
  1, 1;
  1, 2, 1;
  0, 2, 3, 1;
  0, 0, 3, 4, 1;
  0, 0, 0, 4, 5, 1;
  ... (End)

Alois P. Heinz, Rows n = 0..140, flattened
M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 11.
Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 15 Feb. 2013, pp. 1667-1677.
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.
Aoife Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16.
J. Riordan, Letter, Oct 31 1977. The array is on the second page.

First column gives A000110, second column = A005493.
Third column = A003128, row sums = A001861, A059340.
See A244489 for another version of this triangle.
Cf. A059098, A245109, A046716.

a:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, a(n-1, k-1) +(k+1)*(a(n-1, k) +a(n-1, k+1))))
    end:
seq(seq(a(n, k), k=0..n), n=0..15);  # Alois P. Heinz, Nov 30 2012

a[n_, k_] = Sum[StirlingS2[n, i]*Binomial[i, k], {i, 0, n}]; Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]]
(* Jean-François Alcover, Aug 29 2011, after Vladeta Jovovic *)

T(n,k)=if(k<0 || k>n,0,n!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n),k))

def A049020_triangle(dim):
    M = matrix(ZZ, dim, dim)
    for n in (0..dim-1): M[n, n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k]+(k+1)*M[n-1, k+1]
    return M
A049020_triangle(9) # Peter Luschny, Sep 19 2012

a(n,k) = a(n-1, k-1) + (k+1)*a(n-1, k) + (k+1)*a(n-1, k+1), n >= 1.
a(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(i,k). - Vladeta Jovovic, Jan 27 2001
E.g.f. for the k-th column is (1/k!)*(exp(x)-1)^k*exp(exp(x)-1). - Vladeta Jovovic, Jan 27 2001
G.f.: 1/(1-x-xy-x^2(1+y)/(1-2x-xy-2x^2(1+y)/(1-3x-xy-3x^2(1+y)/(1-4x-xy-4x^2(1+y)/(1-... (continued fraction). - Paul Barry, Apr 29 2009
E.g.f.: exp((y+1)*(exp(x)-1)). - Geoffrey Critzer, Nov 30 2012
Note that A244489 (which is essentially the same triangle) has the formula T(n,k) = Sum_{j=k..n} binomial(n,j)*Stirling_2(j,k)*Bell(n-j), where Bell(n) = A000110(n), for n >= 1, 0 <= k <= n-1. - N. J. A. Sloane, May 17 2016
a(2n,n) = A245109(n). - Alois P. Heinz, Aug 23 2017
Empirical: a(n,k) = p(1^n)[st(1^k)] (see A002872 for notation). - Andrey Zabolotskiy, Oct 17 2017
a(n,k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j)/k!. - Peter Luschny, Dec 06 2023

More terms from James Sellers.

Better definition from Geoffrey Critzer, Nov 30 2012.

A059364 Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.

Original entry on oeis.org

1, 2, 1, 6, 7, 2, 24, 46, 29, 6, 120, 326, 329, 146, 24, 720, 2556, 3604, 2521, 874, 120, 5040, 22212, 40564, 39271, 21244, 6084, 720, 40320, 212976, 479996, 598116, 444849, 197380, 48348, 5040, 362880, 2239344, 6023772, 9223012, 8788569

Offset: 1

Vladeta Jovovic, Jan 28 2001

Sum_{k=0..n-1} T(n,k)=(2*n-1)!!.
Alternating row sums = 1. - Gerald McGarvey, Aug 06 2006
Essentially triangle given by [1,1,2,2,3,3,4,4,5,5,6,6,...] DELTA [0,1,1,2,2,3,3,4,4,5,5,...] = [1;1,0;2,1,0;6,7,2,0;24,46,29,6,0;...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 20 2006

			[1],
[2, 1],
[6, 7, 2],
[24, 46, 29, 6],
[120, 326, 329, 146, 24],
[720, 2556, 3604, 2521, 874, 120], ...
2+1=3!!, 6+7+2=5!!, 24+46+29+6=7!!, 120+326+329+146+24=9!!.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Grzegorz Rzadkowski and M. Urlinska, A Generalization of the Eulerian Numbers, arXiv preprint arXiv:1612.06635 [math.CO], 2016-2017.

Cf. A001147, A059340.

Table[Sum[Abs[StirlingS1[n, n - j]]*Binomial[j, k], {j, 0, n}], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* G. C. Greubel, Jan 08 2017 *)

T(n,k)=if(n<1,0,n!*polcoeff(polcoeff((1-x-x*y+x*O(x^n))^(-1/(1+y)),n),k))

def A059364(n,k): return add(stirling_number1(n,n-i)*binomial(i,k) for i in (0..n))
for n in (1..5): [A059364(n,k) for k in (0..n-1)]  # Peter Luschny, May 12 2013

For n>1, T(n,k) = (n-1)*T(n-1,k-1) + n*T(n-1,k) (assuming any T(i,j) outside the triangle = 0). - Gerald McGarvey, Aug 06 2006

cp's OEIS Frontend

A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

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A059605 a(n) = (1/3!)(n^3 + 24n^2 + 107*n + 90), compare A059604.

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A059364 Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.

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cp's OEIS Frontend

A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions

A059605 a(n) = (1/3!)*(n^3 + 24*n^2 + 107*n + 90), compare A059604.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Formula

Extensions

A059364 Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A059605 a(n) = (1/3!)(n^3 + 24n^2 + 107*n + 90), compare A059604.