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Search: id:A011971
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| A011971 |
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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). |
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38
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| 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; listen)
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OFFSET
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0,3
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COMMENT
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Also called the Bell triangle or the Pierce triangle.
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.
a(n,k) is the number of equivalence relations on {1, ...,n} such that 1 not equivalent to k, ..., k-1 not equivalent to k. - D. E. Knuth, Sep 21, 2002.
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REFERENCES
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A. C. Aitken, A problem on combinations, Edinburgh Math. Notes 28 (1933), 18-33.
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, TAOCP, Vol. 4, Section 7.2.1.5.
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).
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LINKS
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T. D. Noe, Rows n=0..50 of triangle, flattened
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Charles Sanders Peirce, Works
Charles Sanders Peirce, Collected Papers
Eric Weisstein's World of Mathematics, Bell Triangle
Nick Hobson, Python program for this sequence
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FORMULA
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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]
a(n,k) = Sum_{i=0..k} binomial(k,i)*Bell(n-k+i). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 15 2006
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EXAMPLE
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Triangle begins:
1;
1,2;
2,3,5;
5,7,10,15;
15,20,27,37,52;
...
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MAPLE
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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:
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MATHEMATICA
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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)
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CROSSREFS
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Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, etc., A011968, A011969. Cf. A046934, A011972 (uniqued).
Main diagonal is in A094577. Mirror image is in A123346.
See also A095149, A106436, A108041, A108042, A108043.
Sequence in context: A033189 A008507 A028364 this_sequence A060048 A110699 A035537
Adjacent sequences: A011968 A011969 A011970 this_sequence A011972 A011973 A011974
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KEYWORD
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tabl,nonn,easy,nice
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AUTHOR
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njas, J. H. Conway and R. K. Guy
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EXTENSIONS
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Peirce reference from Jon Awbrey (jawbrey(AT)att.net), Mar 11, 2002
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