This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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). 53
 1, 1, 2, 2, 3, 5, 5, 7, 10, 15, 15, 20, 27, 37, 52, 52, 67, 87, 114, 151, 203, 203, 255, 322, 409, 523, 674, 877, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147, 21147, 25287, 30304 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also called the Bell triangle or the Peirce triangle. 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. - Don Knuth, Sep 21 2002 [Comment revised by Thijs van Ommen (thijsvanommen(AT)gmail.com), Jul 13 2008] REFERENCES A. C. Aitken, A problem on combinations, Edinburgh Math. Notes 28 (1933), xviii-xxxiii. [This journal is now available online] J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 205. L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 212. D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 418). 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). Jeffrey Shallit, A triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71. LINKS T. D. Noe and Chai Wah Wu, Rows n=0..200 of triangle, flattened (rows n=0..50 from T. D. Noe) H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. See Table IV. H. W. Becker, Rooks and rhymes, Math. Mag., 22 (1948/49), 23-26. [Annotated scanned copy] A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see array p. 228). Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy] D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78. R. K. Guy, Letters to N. J. A. Sloane, June-August 1968 Nick Hobson, Python program for this sequence Don Knuth, Email to N. J. A. Sloane, Jan 29 2018 C. S. Peirce, Assorted Papers C. S. Peirce, Collected Papers C. S. Peirce, Published Works Jeffrey Shallit, A triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71. T. Tichenor, Bounds on graph compositions and the connection to the Bell triangle, Discr. Math., 339 (2016), 1419-1423. Eric Weisstein's World of Mathematics, Bell Triangle D. Wuilquin, Letters to N. J. A. Sloane, August 1984 FORMULA Double-exponential generating function: sum_{n, k} a(n-k, k) x^n y^k / n! k! = exp(e^{x+y}-1+x). - Don Knuth, Sep 21 2002 [U coordinates, reversed] a(n,k) = Sum_{i=0..k} binomial(k,i)*Bell(n-k+i). - Vladeta Jovovic, Oct 15 2006 EXAMPLE Triangle begins: 00:       1 01:       1      2 02:       2      3      5 03:       5      7     10     15 04:      15     20     27     37     52 05:      52     67     87    114    151    203 06:     203    255    322    409    523    674    877 07:     877   1080   1335   1657   2066   2589   3263   4140 08:    4140   5017   6097   7432   9089  11155  13744  17007  21147 09:   21147  25287  30304  36401  43833  52922  64077  77821  94828 115975 10:  115975 137122 162409 192713 229114 272947 325869 389946 467767 562595 678570 ... MAPLE 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; for n from 0 to 12 do lprint([ seq(A011971(n, k), k=0..n) ]); od: MATHEMATICA 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}]] (Robert G. Wilson v, Mar 27 2004) Flatten[Table[Sum[Binomial[k, i]*BellB[n-k+i], {i, 0, k}], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, May 24 2016, after Vladeta Jovovic *) PROG (Haskell) a011971 n k = a011971_tabl !! n !! k a011971_row n = a011971_tabl !! n a011971_tabl = iterate (\row -> scanl (+) (last row) row) [1] -- Reinhard Zumkeller, Dec 09 2012 (Python) # requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs. from itertools import accumulate A011971 = blist = [1] for _ in range(10**2): ....b = blist[-1] ....blist = list(accumulate([b]+blist)) ....A011971 += blist # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 19 2014 CROSSREFS Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, etc., A011968, A011969. Cf. A046934, A011972 (duplicates removed). Main diagonal is in A094577. Mirror image is in A123346. See also A095149, A106436, A108041, A108042, A108043. Sequence in context: A028364 A239482 A280470 * A060048 A110699 A035537 Adjacent sequences:  A011968 A011969 A011970 * A011972 A011973 A011974 KEYWORD tabl,nonn,easy,nice AUTHOR EXTENSIONS Peirce reference from Jon Awbrey, Mar 11 2002 Reference to my paper from Jeffrey Shallit, Jan 23 2015 Moved a comment to A056857 where it belonged. - N. J. A. Sloane, May 02 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 17 07:10 EDT 2018. Contains 300550 sequences. (Running on oeis4.)