A066633 Triangle T(n,k), n >= 1, 1 <= k <= n, giving number of k's in all partitions of n.

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

Offset

1,2

Comments

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

References

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

Links

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

Formula

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

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, 3, 2, 1, 1;

Maple

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

Mathematica

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 *)

Crossrefs

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

Keyword

easy,nonn,tabl

Author

Naohiro Nomoto, Jan 09 2002

Extensions

More terms from Vladeta Jovovic, Jan 11 2002

Status

approved