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A035206
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Number of multisets associated with least integer of each prime signature.
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7
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1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1, 7, 42, 42, 42, 105, 210, 105, 105, 140, 420, 140, 105, 210, 42, 1, 8, 56, 56, 56, 28, 168, 336, 336, 168, 168, 280, 840, 420, 840, 70, 280, 1120, 560, 168, 420, 56, 1, 9, 72
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n,k) multiplied by A036038(n,k) yields A049009(n,k).
a(n,k) enumerates distributions of n identical objects (balls) into m of alltogether n distinguishable boxes. The k-th partition of n, taken in the Abramowitz-Stegun (A-St) order, specifies the occupation of the m =m(n,k)= A036043(n,k) boxes. m = m(n,k) is the number of parts of the k-th partition of n. For the A-St ordering see pp.831-2 of the reference given in A117506. W. Lang, Nov 13 2007.
The sequence of row lengths is p(n)= A000041(n) (partition numbers).
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
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LINKS
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.
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FORMULA
| a(n,k) = A048996(n,k)*binomial(n,m(n,k)),n>=1, k=1,...,p(n) and m(n,k):=A036043(n,k) gives the number of parts of the k-th partition of n.
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EXAMPLE
| 1;
2,1;
3,6,1;
4,12,6,12,1;
5,20,20,30,30,20,1;
...
a(5,5) relates to the partition (1,2^2) of n=5. Here m=3 and 5 indistinguishable (identical)
balls are put into boxes b1,...,b5 with m=3 boxes occupied; one with one ball and two with two balls.
Therefore a(5,5) = binomial(5,3)*3!/(1!*2!) = 10*3 = 30. W. Lang, Nov 13 2007.
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CROSSREFS
| Cf. A036038, A048996, A049009.
Cf. A001700 (row sums).
Cf. A103371(n-1, m-1) (triangle obtained after summing in every row the numbers with like part numbers m).
Sequence in context: A078760 A103280 A046899 * A181511 A115196 A093346
Adjacent sequences: A035203 A035204 A035205 * A035207 A035208 A035209
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KEYWORD
| nonn,tabf,easy
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AUTHOR
| Alford Arnold (Alford1940(AT)aol.com)
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EXTENSIONS
| More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jul 27 2006
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