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Search: id:A072233
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| A072233 |
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Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty. |
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1
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| 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1, 0, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 0, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 0, 1, 6, 14, 18, 18, 14, 11, 7
(list; table; graph; listen)
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OFFSET
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0,13
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COMMENT
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Regarded as a triangular table, this is another version of the number of partitions of n into k parts, A008284. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Dec 18 2006
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LINKS
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COS, Information on Numerical Partitions
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FORMULA
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T(0, k) = 1, T(n, 0) = 0 (n>0), T(1, k) = 1 (k>0), T(n, 1) = 1 (n>0), T(n, k) = 0 for n < 0, T(n, k) = Sum[ T(n-k+i, k-i), i=0...k-1] Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k>n), T(n, k) = T(n-1, k-1) + T(n-k, k).
G.f. Product_{j=0}^{infinity} 1/(1-xy^j). Regarded as a triangular array, g.f. Product_{j=1}^{infinity} 1/(1-xy^j). - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Dec 18 2006
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EXAMPLE
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Table begins (upper left corner = T(0,0)):
1 1 1 1 1 1 1 1 1 ...
0 1 1 1 1 1 1 1 1 ...
0 1 2 2 2 2 2 2 2 ...
0 1 2 3 3 3 3 3 3 ...
0 1 3 4 5 5 5 5 5 ...
0 1 3 5 6 7 7 7 7 ...
0 1 4 7 9 10 11 11 11 ...
0 1 4 8 11 13 14 15 15 ...
0 1 5 10 15 18 20 21 22 ...
There is 1 way to distribute 0 objects into k containers: T(0, k) = 1. The different ways for n=4, k=3 are: (oooo)()(), (ooo)(o)(), (oo)(oo)(), (oo)(o)(o), so T(4, 3) = 4.
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CROSSREFS
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Sum of antidiagonal entries T(n, k) with n+k=m equals A000041(m).
Cf. A008284.
Sequence in context: A054078 A029400 A069713 this_sequence A116598 A068914 A090824
Adjacent sequences: A072230 A072231 A072232 this_sequence A072234 A072235 A072236
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Martin Wohlgemuth (mail(AT)matroid.com), Jul 05 2002
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EXTENSIONS
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Corrected by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Dec 18 2006
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