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A049020 Triangle of numbers a(n,k), 0<=k<=n, related to Bell numbers. 15
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

a(n) is the number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished. Row sums give A001861. - Geoffrey Critzer, Nov 30 2012

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.

P. Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.

J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16.

FORMULA

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

EXAMPLE

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)

MAPLE

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);

MATHEMATICA

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 V. Jovovic *)

PROG

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

(Sage)

def A049020_triangle(dim):

    M = matrix(SR, 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

CROSSREFS

First column gives A000110, second column = A005493.

Third column = A003128, row sums = A001861, A059340.

See A244489 for another version of this triangle.

Sequence in context: A060693 A172381 A089302 * A144634 A178125 A147315

Adjacent sequences:  A049017 A049018 A049019 * A049021 A049022 A049023

KEYWORD

nonn,tabl,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

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Last modified September 1 00:54 EDT 2016. Contains 276001 sequences.