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

%I A066633 #108 Feb 16 2025 08:32:45
%S A066633 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,
%T A066633 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,
%U A066633 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
%N A066633 Triangle T(n,k), n >= 1, 1 <= k <= n, giving number of k's in all partitions of n.
%C A066633 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
%C A066633 It appears that reversed rows converge to A000041. - _Omar E. Pol_, Mar 11 2012
%C A066633 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
%C A066633 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
%D A066633 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
%H A066633 Alois P. Heinz, <a href="/A066633/b066633.txt">Rows n = 1..141, flattened</a>
%H A066633 Manosij Ghosh Dastidar and Sourav Sen Gupta, <a href="https://arxiv.org/abs/1111.0094">Generalization of a few results in Integer Partitions</a>, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
%H A066633 Joseph Vandehey, <a href="https://math.colgate.edu/~integers/a18Proc23/a18Proc23.pdf">Digital problems in the theory of partitions</a>, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.
%H A066633 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EldersTheorem.html">Elder's Theorem</a>
%F A066633 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
%F A066633 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
%F A066633 Equals triangle A027293 * A051731 as infinite lower triangular matrices. - _Gary W. Adamson_ Mar 21 2011
%F A066633 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
%F A066633 It appears that T(n,k) = A206563(n,k) - A206563(n,k+2). - _Omar E. Pol_, Feb 26 2012
%F A066633 T(n,k) = Sum_{j=1..n} A182703(j,k). - _Omar E. Pol_, May 02 2012
%e A066633 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.
%e A066633 Triangle begins:
%e A066633     1;
%e A066633     2,   1;
%e A066633     4,   1,  1;
%e A066633     7,   3,  1,  1;
%e A066633    12,   4,  2,  1,  1;
%e A066633    19,   8,  4,  2,  1,  1;
%e A066633    30,  11,  6,  3,  2,  1,  1;
%e A066633    45,  19,  9,  6,  3,  2,  1, 1;
%e A066633    67,  26, 15,  8,  5,  3,  2, 1, 1;
%e A066633    97,  41, 21, 13,  8,  5,  3, 2, 1, 1;
%e A066633   139,  56, 31, 18, 12,  7,  5, 3, 2, 1, 1;
%e A066633   195,  83, 45, 28, 17, 12,  7, 5, 3, 2, 1, 1;
%e A066633   272, 112, 63, 38, 25, 16, 11, 7, 5, 3, 2, 1, 1;
%e A066633   ...
%p A066633 b:= proc(n, i) option remember;
%p A066633       `if`(n=0 or i=1, 1+n*x, b(n, i-1)+
%p A066633       `if`(i>n, 0, (g->g+coeff(g, x, 0)*x^i)(b(n-i, i))))
%p A066633     end:
%p A066633 T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n, n)):
%p A066633 seq(T(n), n=1..14);  # _Alois P. Heinz_, Mar 21 2012
%t A066633 Table[Count[Flatten[IntegerPartitions[n]],k],
%t A066633 {n,1,20},{k,1,n}]
%t A066633 TableForm[% ] (* as a triangle *)
%t A066633 Flatten[%%]   (* as a sequence *)
%t A066633 (* _Clark Kimberling_, Mar 03 2010 *)
%t A066633 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_ *)
%o A066633 (Python)
%o A066633 from math import isqrt, comb
%o A066633 from sympy import partition
%o A066633 def A066633(n):
%o A066633     a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
%o A066633     b = n-comb(a,2)
%o A066633     return sum(partition(j) for j in range(a%b,a,b)) # _Chai Wah Wu_, Nov 13 2024
%Y A066633 Row sums give positive terms of A006128.
%Y A066633 Columns (1-10): A000070, A024786-A024794.
%K A066633 easy,nonn,tabl
%O A066633 1,2
%A A066633 _Naohiro Nomoto_, Jan 09 2002
%E A066633 More terms from _Vladeta Jovovic_, Jan 11 2002