A365434 -id:A365434 - cp's OEIS frontend

A385001 Irregular triangle read by rows: T(n,k) is the number of partitions of n with k designated summands, n >= 0, 0 <= k <= A003056(n).

1, 0, 1, 0, 3, 0, 4, 1, 0, 7, 3, 0, 6, 9, 0, 12, 15, 1, 0, 8, 30, 3, 0, 15, 45, 9, 0, 13, 67, 22, 0, 18, 99, 42, 1, 0, 12, 135, 81, 3, 0, 28, 175, 140, 9, 0, 14, 231, 231, 22, 0, 24, 306, 351, 51, 0, 24, 354, 551, 97, 1, 0, 31, 465, 783, 188, 3, 0, 18, 540, 1134, 330, 9

Offset: 0

Omar E. Pol, Jul 17 2025

The divisor function sigma_1(n) = A000203(n) is also the number of partitions of n with only one designated summand, n >= 1.
When part i has multiplicity j > 0 exactly one part i is "designated".
The length of the row n is A002024(n+1) = 1 + A003056(n), hence the first positive element in column k is in the row A000217(k).
Alternating row sums give A329157.
Columns converge to A000716.
This triangle equals A060043 with reversed rows and an additional column 0.

			Triangle begins:
--------------------------------------------
   n\k:   0    1     2     3     4    5   6
--------------------------------------------
   0 |    1;
   1 |    0,   1;
   2 |    0,   3;
   3 |    0,   4,    1;
   4 |    0,   7,    3;
   5 |    0,   6,    9;
   6 |    0,  12,   15,    1;
   7 |    0,   8,   30,    3;
   8 |    0,  15,   45,    9;
   9 |    0,  13,   67,   22;
  10 |    0,  18,   99,   42,    1;
  11 |    0,  12,  135,   81,    3;
  12 |    0,  28,  175,  140,    9;
  13 |    0,  14,  231,  231,   22;
  14 |    0,  24,  306,  351,   51;
  15 |    0,  24,  354,  551,   97,   1;
  16 |    0,  31,  465,  783,  188,   3;
  17 |    0,  18,  540, 1134,  330,   9;
  18 |    0,  39,  681, 1546,  568,  22;
  19 |    0,  20,  765, 2142,  918,  51;
  20 |    0,  42,  945, 2835, 1452, 108;
  21 |    0,  32, 1040, 3758, 2233, 208,  1;
  ...
For n = 6 and k = 1 there are 12 partitions of 6 with only one designated summand as shown below:
   6'
   3'+ 3
   3 + 3'
   2'+ 2 + 2
   2 + 2'+ 2
   2 + 2 + 2'
   1'+ 1 + 1 + 1 + 1 + 1
   1 + 1'+ 1 + 1 + 1 + 1
   1 + 1 + 1'+ 1 + 1 + 1
   1 + 1 + 1 + 1'+ 1 + 1
   1 + 1 + 1 + 1 + 1'+ 1
   1 + 1 + 1 + 1 + 1 + 1'
So T(6,1) = 12, the same as A000203(6) = 12.
.
For n = 6 and k = 2 there are 15 partitions of 6 with two designated summands as shown below:
   5'+ 1'
   4'+ 2'
   4'+ 1'+ 1
   4'+ 1 + 1'
   3'+ 1'+ 1 + 1
   3'+ 1 + 1'+ 1
   3'+ 1 + 1 + 1'
   2'+ 2 + 1'+ 1
   2'+ 2 + 1 + 1'
   2 + 2'+ 1'+ 1
   2 + 2'+ 1 + 1'
   2'+ 1'+ 1 + 1 + 1
   2'+ 1 + 1'+ 1 + 1
   2'+ 1 + 1 + 1'+ 1
   2'+ 1 + 1 + 1 + 1'
So T(6,2) = 15, the same as A002127(6) = 15.
.
For n = 6 and k = 3 there is only one partition of 6 with three designated summands as shown below:
   3'+ 2'+ 1'
So T(6,3) = 1, the same as A002128(6) = 1.
There are 28 partitions of 6 with designated summands, so A077285(6) = 28.
.

Alois P. Heinz, Rows n = 0..1000, flattened

Columns: A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5), A384926 (k=6).
Row sums give A077285.
Cf. A000217, A002024, A003056, A060043, A196020, A252117, A329157, A365434, A384999.
Cf. A000096, A000716, A116608, A293421, A384998, A384999.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 18 2025

From Alois P. Heinz, Jul 18 2025: (Start)
Sum_{k>=1} k * T(n,k) = A293421(n).
T(A000096(n),n) = A000716(n). (End)
G.f.: Product_{i>0} 1 + (y*x^i)/(1 - x^i)^2. - John Tyler Rascoe, Jul 23 2025
Conjecture: For fixed k >= 1, Sum_{j=1..n} T(j,k) ~ Pi^(2*k) * n^(2*k) / ((2*k)! * (2*k+1)!). - Vaclav Kotesovec, Aug 01 2025

A365195 Height of the first staircase of the ziggurat of order n described in A347186.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 8, 5, 9, 6, 12, 7, 12, 8, 16, 9, 18, 10, 20, 11, 18, 12, 24, 13, 21, 14, 28, 15, 30, 16, 32, 17, 27, 18, 36, 19, 30, 20, 40, 21, 42, 22, 42, 23, 36, 24, 48, 25, 39, 26, 49, 27, 54, 28, 56, 29, 45, 30, 60, 31, 48, 32, 64, 33, 66, 34, 63, 35, 54, 36, 72, 37, 57, 38, 70, 39, 77, 40, 80

Offset: 1

Omar E. Pol, Aug 25 2023

nonn

a(n) is also the number of square cells in the first layer of the symmetric representation of sigma(n) from the border to, at most, the axis of symmetry of the diagram.
The values of a(n) where a(n) = n give A174973.
Since this is a supersequence of A174973 so all powers of 2 and all even perfect numbers are in the sequence.
From Omar E. Pol, Oct 22 2023: (Start)
The values of a(n) where a(n) is not equal to n give A238524.
If n is an odd prime then a(n) = (n + 1)/2.
Shares infinitely many terms with A365433 from which first differs at a(15). (End)

Cf. A000079, A000396, A065091, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238524, A262626, A279387, A347186, A347367, A365433, A365434.

a(n) <= n.

a(2*n-1) = n.

More terms from Omar E. Pol, Oct 22 2023

A365433 Height of the ziggurat of order n described in A347186.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 8, 5, 9, 6, 12, 7, 12, 10, 16, 9, 18, 10, 20, 13, 18, 12, 24, 13, 21, 16, 28, 15, 30, 16, 32, 19, 27, 21, 36, 19, 30, 22, 40, 21, 42, 22, 42, 40, 36, 24, 48, 25, 39, 28, 49, 27, 54, 36, 56, 31, 45, 30, 60, 31, 48, 36, 64, 41, 66, 34, 63, 37, 56, 36, 72, 37, 57, 61, 70, 50, 77, 40, 80

Offset: 1

Omar E. Pol, Oct 18 2023

nonn

The values of a(n) where a(n) = n give A174973.
The values of a(n) where a(n) is not equal to n give A238524.
If n is an odd prime then a(n) = (n + 1)/2.
Since this is a supersequence of A174973 so all powers of 2 and all even perfect numbers are in the sequence.
Shares infinitely many terms with A365195 from which first differs at a(15).

Row lengths of A365434.

Cf. A000079, A000396, A065091, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238524, A245092, A262626, A347186, A347263, A347367, A365195.

a(2*n-1) >= n.

cp's OEIS Frontend

A385001 Irregular triangle read by rows: T(n,k) is the number of partitions of n with k designated summands, n >= 0, 0 <= k <= A003056(n).

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A365195 Height of the first staircase of the ziggurat of order n described in A347186.

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A365433 Height of the ziggurat of order n described in A347186.

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