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A049020 Triangle of numbers a(n,k), 0<=k<=n, related to Bell numbers. 14
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

REFERENCES

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.

W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16. - N. J. A. Sloane, Feb 07 2009

LINKS

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

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) ). 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

EXAMPLE

Triangle begins:

1;

1,1;

2,3,1;

5,10,6,1;

15,37,31,10,1;

...

Contribution 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.

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 July 29 08:48 EDT 2014. Contains 245020 sequences.