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A066633
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Triangle T(n,k), n>=1, 1<=k<=n, giving number of k's in all partitions of n.
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23
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1, 2, 1, 4, 1, 1, 7, 3, 1, 1, 12, 4, 2, 1, 1, 19, 8, 4, 2, 1, 1, 30, 11, 6, 3, 2, 1, 1, 45, 19, 9, 6, 3, 2, 1, 1, 67, 26, 15, 8, 5, 3, 2, 1, 1, 97, 41, 21, 13, 8, 5, 3, 2, 1, 1, 139, 56, 31, 18, 12, 7, 5, 3, 2, 1, 1, 195, 83, 45, 28, 17, 12, 7, 5, 3, 2, 1, 1, 272, 112, 63, 38, 25, 16, 11, 7, 5
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, Arxiv preprint arXiv:1111.0094, 2011
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LINKS
| Eric Weisstein's World of Mathematics, Elder's Theorem
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FORMULA
| T(n, 1) + ... + T(n, n) = A006128(n).
G.f. for the number of k's in all partitions of n is x^k/(1-x^k)*Product_{m>=1} 1/(1-x^m). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 15 2002
T(n, k) = Sum_{j<n, j=n (mod k)} P(j), P(j) = number of partitions of j, P(0) = 1 - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
Equals triangle A027293 * A051731 as infinite lower triangular matrices. [Gary W. Adamson (qntmpkt(AT)yahoo.com) Mar 21 2011]
It appears that T(n+k,k) = T(n,k) + A000041(n). - Omar E. Pol, Feb 04 2012
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EXAMPLE
| For n = 3, k = 1; 3 = 2+1 = 1+1+1. T(3,1) = (zero 1's) + (one 1's) + (three 1's), so T(3,1) = 4.
Triangle begins
1,
2, 1,
4, 1, 1,
7, 3, 1, 1,
12, 4, 2, 1, 1,
19, 8, 4, 2, 1, 1,
30, 11, 6, 3, 2, 1, 1,
45, 19, 9, 6, 3, 2, 1, 1,
67, 26, 15, 8, 5, 3, 2, 1, 1,
97, 41, 21, 13, 8, 5, 3, 2, 1, 1,
139, 56, 31, 18, 12, 7, 5, 3, 2, 1, 1,
195, 83, 45, 28, 17, 12, 7, 5, 3, 2, 1, 1,
272, 112, 63, 38, 25, 16, 11, 7, 5, ...
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CROSSREFS
| Diagonals: A000070, A024786-A024794.
Sequence in context: A191314 A191306 A191525 * A088443 A117352 A137710
Adjacent sequences: A066630 A066631 A066632 * A066634 A066635 A066636
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KEYWORD
| easy,nonn,tabl,changed
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AUTHOR
| Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 09 2002
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 11 2002
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