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%I
%S 1,1,2,2,3,5,5,7,10,15,15,20,27,37,52,52,67,87,114,151,203,203,255,
%T 322,409,523,674,877,877,1080,1335,1657,2066,2589,3263,4140,4140,5017,
%U 6097,7432,9089,11155,13744,17007,21147,21147,25287,30304
%N Aitken's array: triangle of numbers {a(n,k), n >= 0, 0<=k<=n} read by rows, defined by a(0,0)=1, a(n,0)=a(n-1,n-1), a(n,k)=a(n,k-1)+a(n-1,k-1).
%C Also called the Bell triangle or the Peirce triangle.
%C Let P be the lower-triangular Pascal-matrix, Then this is exp(P) / exp(1). - _Gottfried Helms_, Mar 30 2007.
%C a(n,k) is the number of equivalence relations on {0, ..., n} such that k is not equivalent to n, k+1 is not equivalent to n, ..., n-1 is not equivalent to n. - D. E. Knuth, Sep 21, 2002. [Comment revised by Thijs van Ommen (thijsvanommen(AT)gmail.com), Jul 13 2008]
%D A. C. Aitken, A problem on combinations, Edinburgh Math. Notes 28 (1933), 18-33.
%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 205.
%D Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; Mathematical Notes: On the Number of Partitionings of a Set of n Distinct Objects. Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841 - From _N. J. A. Sloane_, Jul 31 2012
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 212.
%D D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 418).
%D Charles Sanders Peirce, On the Algebra of Logic, American Journal of Mathematics, Vol. 3, pages 15-57, 1880. Reprinted in Collected Papers (1935-1958) and in Writings of Charles S. Peirce: A Chronological Edition (Indiana University Press, Bloomington, IN, 1986).
%H T. D. Noe, <a href="/A011971/b011971.txt">Rows n=0..50 of triangle, flattened</a>
%H D. Dumont, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05dumont.html">Matrices d'Euler-Seidel</a>, Sem. Loth. Comb. B05c (1981) 59-78.
%H Charles Sanders Peirce, <a href="http://members.door.net/arisbe/menu/library/bycsp/bycsp.htm">Works</a> [broken link]
%H Charles Sanders Peirce, <a href="http://www.nlx.com/authors/123">Collected Papers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellTriangle.html">Bell Triangle</a>
%H Nick Hobson, <a href="/A011971/a011971.py.txt">Python program for this sequence</a>
%F Double-exponential generating function: sum_{n, k} a(n-k, k) x^n y^k / n! k! = exp(e^{x+y}-1+x). - D. E. Knuth, Sep 21, 2002. [U coordinates, reversed]
%F a(n,k) = Sum_{i=0..k} binomial(k,i)*Bell(n-k+i). - _Vladeta Jovovic_, Oct 15 2006
%e Triangle begins:
%e 1;
%e 1,2;
%e 2,3,5;
%e 5,7,10,15;
%e 15,20,27,37,52;
%e ...
%p A011971 := proc(n,k) option remember; if n=0 and k=0 then 1 elif k=0 then A011971(n-1,n-1) else A011971(n,k-1)+A011971(n-1,k-1); fi: end;
%p for n from 0 to 12 do lprint([ seq(A011971(n,k),k=0..n) ]); od:
%t a[0, 0] = 1; a[n_, 0] := a[n - 1, n - 1]; a[n_, k_] := a[n, k - 1] + a[n - 1, k - 1]; Flatten[ Table[ a[n, k], {n, 0, 9}, {k, 0, n}]] (from Robert G. Wilson v Mar 27 2004)
%o (Haskell)
%o a011971 n k = a011971_tabl !! n !! k
%o a011971_row n = a011971_tabl !! n
%o a011971_tabl = iterate (\row -> scanl (+) (last row) row) [1]
%o -- _Reinhard Zumkeller_, Dec 09 2012
%Y Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, etc., A011968, A011969. Cf. A046934, A011972 (duplicates removed).
%Y Main diagonal is in A094577. Mirror image is in A123346.
%Y See also A095149, A106436, A108041, A108042, A108043.
%K tabl,nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, J. H. Conway and R. K. Guy
%E Peirce reference from _Jon Awbrey_, Mar 11, 2002
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