%I A011971
%S A011971 1,1,2,2,3,5,5,7,10,15,15,20,27,37,52,52,67,87,114,151,203,203,255,
%T A011971 322,409,523,674,877,877,1080,1335,1657,2066,2589,3263,4140,4140,
%U A011971 5017,6097,7432,9089,11155,13744,17007,21147,21147,25287,30304
%N A011971 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 A011971 Also called the Bell triangle or the Pierce triangle.
%C A011971 Let P be the lower-triangular Pascal-matrix, Then this is exp(P) / exp(1). 
               - Gottfried Helms (helms(AT)uni-kassel.de), Mar 30 2007.
%C A011971 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 A011971 A. C. Aitken, A problem on combinations, Edinburgh Math. Notes 28 (1933), 
               18-33.
%D A011971 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 
               2003, p. 205.
%D A011971 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 212.
%D A011971 D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.5.
%D A011971 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 A011971 T. D. Noe, <a href="b011971.txt">Rows n=0..50 of triangle, flattened</
               a>
%H A011971 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 A011971 Charles Sanders Peirce, <a href="http://members.door.net/arisbe/menu/
               library/bycsp/bycsp.htm">Works</a>
%H A011971 Charles Sanders Peirce, <a href="http://www.nlx.com/titles/titlpeir.htm">
               Collected Papers</a>
%H A011971 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BellTriangle.html">Bell Triangle</a>
%H A011971 Nick Hobson, <a href="a011971.py.txt">Python program for this sequence</
               a>
%F A011971 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 A011971 a(n,k) = Sum_{i=0..k} binomial(k,i)*Bell(n-k+i). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Oct 15 2006
%e A011971 Triangle begins:
%e A011971 1;
%e A011971 1,2;
%e A011971 2,3,5;
%e A011971 5,7,10,15;
%e A011971 15,20,27,37,52;
%e A011971 ...
%p A011971 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 A011971 for n from 0 to 12 do lprint([ seq(A011971(n,k),k=0..n) ]); od:
%t A011971 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)
%Y A011971 Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, 
               etc., A011968, A011969. Cf. A046934, A011972 (duplicates removed).
%Y A011971 Main diagonal is in A094577. Mirror image is in A123346.
%Y A011971 See also A095149, A106436, A108041, A108042, A108043.
%Y A011971 Sequence in context: A033189 A008507 A028364 this_sequence A060048 A110699 
               A035537
%Y A011971 Adjacent sequences: A011968 A011969 A011970 this_sequence A011972 A011973 
               A011974
%K A011971 tabl,nonn,easy,nice
%O A011971 0,3
%A A011971 N. J. A. Sloane (njas(AT)research.att.com), J. H. Conway and R. K. Guy
%E A011971 Peirce reference from Jon Awbrey (jawbrey(AT)att.net), Mar 11, 2002