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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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68
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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, 3, 2, 1, 1
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OFFSET
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1,2
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COMMENTS
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It appears that row n lists the first differences of the row n of triangle A181187 together with 1 (as the final term of the row n). - Omar E. Pol, Feb 26 2012
It appears that reversed rows converge to A000041. - Omar E. Pol, Mar 11 2012
Proof: For a partition of n with k>floor(n/2+1), k can only occur as the largest part; the other parts sum to n-k, so that T(n,n-k)=A000041(k). - George Beck, Jun 30 2019
T(n,k) is also the total number k's that are divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. - Omar E. Pol, Feb 05 2021
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.2.1.5, Problem 73(b), pp. 415, 761. - N. J. A. Sloane, Dec 30 2018
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LINKS
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Alois P. Heinz, Rows n = 1..141, flattened
Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
Eric Weisstein's World of Mathematics, Elder's Theorem
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FORMULA
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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, 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 Mar 21 2011
It appears that T(n+k,k) = T(n,k) + A000041(n). - Omar E. Pol, Feb 04 2012. This was proved in the Dastidar-Gupta paper in Lemma 1. - George Beck, Jun 26 2019
It appears that T(n,k) = A206563(n,k) - A206563(n,k+2). - Omar E. Pol, Feb 26 2012
T(n,k) = Sum_{j=1..n} A182703(j,k). - Omar E. Pol, May 02 2012
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EXAMPLE
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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, 3, 2, 1, 1;
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MAPLE
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b:= proc(n, i) option remember;
`if`(n=0 or i=1, 1+n*x, b(n, i-1)+
`if`(i>n, 0, (g->g+coeff(g, x, 0)*x^i)(b(n-i, i))))
end:
T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n, n)):
seq(T(n), n=1..14); # Alois P. Heinz, Mar 21 2012
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MATHEMATICA
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Table[Count[Flatten[IntegerPartitions[n]], k],
{n, 1, 20}, {k, 1, n}]
TableForm[% ] (* as a triangle *)
Flatten[%%] (* as a sequence *)
(* Clark Kimberling, Mar 03 2010 *)
T[n_, n_] = 1; T[n_, k_] /; k<n := T[n, k] = T[n-k, k] + PartitionsP[n-k]; T[_, _] = 0; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 21 2015, after Omar E. Pol *)
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CROSSREFS
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Row sums give positive terms of A006128.
Columns (1-10): A000070, A024786-A024794.
Sequence in context: A191314 A191306 A191525 * A238415 A283826 A088443
Adjacent sequences: A066630 A066631 A066632 * A066634 A066635 A066636
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Naohiro Nomoto, Jan 09 2002
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EXTENSIONS
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More terms from Vladeta Jovovic, Jan 11 2002
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STATUS
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approved
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