A123346 Mirror image of the Bell triangle A011971, which is also called the Pierce triangle or Aitken's array.

1, 2, 1, 5, 3, 2, 15, 10, 7, 5, 52, 37, 27, 20, 15, 203, 151, 114, 87, 67, 52, 877, 674, 523, 409, 322, 255, 203, 4140, 3263, 2589, 2066, 1657, 1335, 1080, 877, 21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140, 115975, 94828, 77821, 64077, 52922, 43833, 36401, 30304, 25287, 21147

Offset: 0

Author

N. J. A. Sloane, Oct 14 2006

Keywords

• nonn
• tabl

Comments

a(n,k) is k-th difference of Bell numbers, with a(n,1) = A000110(n) for n>0, a(n,k) = a(n,k-1) - a(n-1, k-1), k<=n, with diagonal (k=n) also equal to Bell numbers (n>=0). - Richard R. Forberg, Jul 13 2013

From Don Knuth, Jan 29 2018: (Start)

If the offset here is changed from 0 to 1, then we can say:

a(n,k) is the number of equivalence classes of [n] in which 1 not equiv to 2, ..., 1 not equiv to k.

In Volume 4A, page 418, I pointed out that a(n,k) is the number of set partitions in which k is the smallest of its block.

And in exercise 7.2.1.5--33, I pointed out that a(n,k) is the number of equivalence relations in which 1 not equiv to 2, 2 not equiv to 3, ..., k-1 not equiv to k. (End)

Examples

			Triangle begins:
    1
    2   1
    5   3   2
   15  10   7  5
   52  37  27 20 15
  203 151 114 87 67 52
  ...
		

References

• D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 418).

Links

• Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
• A. Dil, Veli Kurt, Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices, J. Int. Seq. 14 (2011) # 11.4.6
• Don Knuth, Email to N. J. A. Sloane, Jan 29 2018
• Eric Weisstein's World of Mathematics, Bell Triangle.

Crossrefs

Cf. A011971. Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, A011968, A011969, A046934, A011972, A094577, A095149, A106436, A108041, A108042, A108043.

Programs

Haskell

a123346 n k = a123346_tabl !! n !! k
a123346_row n = a123346_tabl !! n
a123346_tabl = map reverse a011971_tabl
-- Reinhard Zumkeller, Dec 09 2012

Mathematica

a[n_, k_] := Sum[Binomial[n - k, i - k] BellB[i], {i, k, n}];
Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 03 2018 *)

Python

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A123346_list = blist = [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A123346_list += reversed(blist)
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

Formula

a(n,k) = Sum_{i=k..n} binomial(n-k,i-k)*Bell(i). - Vladeta Jovovic, Oct 14 2006

Extensions

More terms from Alexander Adamchuk and Vladeta Jovovic, Oct 14 2006