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

%I A123346 #48 Feb 16 2025 08:33:02
%S A123346 1,2,1,5,3,2,15,10,7,5,52,37,27,20,15,203,151,114,87,67,52,877,674,
%T A123346 523,409,322,255,203,4140,3263,2589,2066,1657,1335,1080,877,21147,
%U A123346 17007,13744,11155,9089,7432,6097,5017,4140,115975,94828,77821,64077,52922,43833,36401,30304,25287,21147
%N A123346 Mirror image of the Bell triangle A011971, which is also called the Pierce triangle or Aitken's array.
%C A123346 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
%C A123346 From _Don Knuth_, Jan 29 2018: (Start)
%C A123346 If the offset here is changed from 0 to 1, then we can say:
%C A123346 a(n,k) is the number of equivalence classes of [n] in which 1 not equiv to 2, ..., 1 not equiv to k.
%C A123346 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.
%C A123346 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)
%D A123346 D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 418).
%H A123346 Reinhard Zumkeller, <a href="/A123346/b123346.txt">Rows n = 0..100 of triangle, flattened</a>
%H A123346 A. Dil, Veli Kurt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Dil/dil5.html">Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices</a>, J. Int. Seq. 14 (2011) # 11.4.6
%H A123346 Don Knuth, <a href="/A040027/a040027.txt">Email to N. J. A. Sloane</a>, Jan 29 2018
%H A123346 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellTriangle.html">Bell Triangle</a>.
%F A123346 a(n,k) = Sum_{i=k..n} binomial(n-k,i-k)*Bell(i). - _Vladeta Jovovic_, Oct 14 2006
%e A123346 Triangle begins:
%e A123346     1
%e A123346     2   1
%e A123346     5   3   2
%e A123346    15  10   7  5
%e A123346    52  37  27 20 15
%e A123346   203 151 114 87 67 52
%e A123346   ...
%t A123346 a[n_, k_] := Sum[Binomial[n - k, i - k] BellB[i], {i, k, n}];
%t A123346 Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 03 2018 *)
%o A123346 (Haskell)
%o A123346 a123346 n k = a123346_tabl !! n !! k
%o A123346 a123346_row n = a123346_tabl !! n
%o A123346 a123346_tabl = map reverse a011971_tabl
%o A123346 -- _Reinhard Zumkeller_, Dec 09 2012
%o A123346 (Python)
%o A123346 # requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
%o A123346 from itertools import accumulate
%o A123346 A123346_list = blist = [1]
%o A123346 for _ in range(2*10**2):
%o A123346     b = blist[-1]
%o A123346     blist = list(accumulate([b]+blist))
%o A123346     A123346_list += reversed(blist)
%o A123346 # _Chai Wah Wu_, Sep 02 2014, updated _Chai Wah Wu_, Sep 20 2014
%Y A123346 Cf. A011971. Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, A011968, A011969, A046934, A011972, A094577, A095149, A106436, A108041, A108042, A108043.
%K A123346 nonn,tabl
%O A123346 0,2
%A A123346 _N. J. A. Sloane_, Oct 14 2006
%E A123346 More terms from _Alexander Adamchuk_ and _Vladeta Jovovic_, Oct 14 2006