cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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).

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Jul 17 2025

Keywords

Comments

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.

Examples

			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.
.

Links

Crossrefs

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.

Programs

  • Maple
    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

Formula

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

A293422 The PDO_t(n) function (Number of tagged parts over all the partitions of n with designated summands in which all parts are odd).

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 24, 36, 52, 74, 104, 144, 196, 264, 352, 468, 614, 800, 1036, 1332, 1704, 2168, 2744, 3456, 4331, 5408, 6724, 8328, 10278, 12640, 15496, 18936, 23072, 28030, 33960, 41040, 49470, 59488, 71368, 85428, 102042, 121632, 144692, 171792, 203584
Offset: 1

Views

Author

Seiichi Manyama, Oct 08 2017

Keywords

Examples

			n = 4                 n = 5                     n = 6
-------------------   -----------------------   ---------------------------
3'+ 1'        -> 2    5'                -> 1    5'+ 1'                -> 2
1'+ 1 + 1 + 1 -> 1    3'+ 1'+ 1         -> 2    3'+ 3                 -> 1
1 + 1'+ 1 + 1 -> 1    3'+ 1 + 1'        -> 2    3 + 3'                -> 1
1 + 1 + 1'+ 1 -> 1    1'+ 1 + 1 + 1 + 1 -> 1    3'+ 1'+ 1 + 1         -> 2
1 + 1 + 1 + 1'-> 1    1 + 1'+ 1 + 1 + 1 -> 1    3'+ 1 + 1'+ 1         -> 2
                      1 + 1 + 1'+ 1 + 1 -> 1    3'+ 1 + 1 + 1'        -> 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 + 1'+ 1 + 1 -> 1
                                                1 + 1 + 1 + 1 + 1'+ 1 -> 1
                                                1 + 1 + 1 + 1 + 1 + 1'-> 1
-------------------   -----------------------   ---------------------------
a(4)          =  6.   a(5)              = 10.   a(6)                  = 16.

Links

Crossrefs

Cf. A102186 (PDO(n)), A293421.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-x^(2*k)) * (1-x^(3*k))^2 * (1-x^(12*k))^2 / ((1-x^k)^2 * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 15 2017 *)
  • Ruby
    def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A(n)
  partition(n, 1, n).select{|i| i.all?{|j| j.odd?}}.map{|a| a.each_with_object(Hash.new(0)){|v, o| o[v] += 1}.values}.map{|i| i.size * i.inject(:*)}.inject(:+)
end
def A293422(n)
  (1..n).map{|i| A(i)}
end
p A293422(40)

Formula

G.f.: q * Product_{k>0} ((1 - q^(2*k))*(1 - q^(3*k))^2*(1 - q^(12*k))^2)/((1 - q^k)^2*(1 - q^(6*k))).
a(n) ~ exp(sqrt(5*n)*Pi/3) / (3 * 2^(3/2) * 5^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 15 2017

A293423 Expansion of Product_{k>0} (1 - q^(3*k))^5/((1 - q^k)^3*(1 - q^(6*k))^2).

Original entry on oeis.org

1, 3, 9, 17, 36, 63, 118, 195, 333, 528, 852, 1305, 2020, 3012, 4518, 6583, 9624, 13761, 19698, 27702, 38952, 54000, 74784, 102357, 139882, 189297, 255690, 342497, 457824, 607617, 804656, 1058970, 1390545, 1815984, 2366268, 3068388, 3970008, 5114382, 6574266
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2017

Keywords

Links

Crossrefs

Cf. A293421.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i>1, b(n, i-1), 0)+
      add((p-> p+[0, p[1]])(b(n-i*j, min(n-i*j, i-1))*j), j=`if`(i=1, n, 1..n/i)))
    end:
a:= n-> (p-> 2*p[2]+p[1])(b(n$2)):
seq(a(n), n=0..38);  # Alois P. Heinz, Jul 18 2025
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))^5 / ((1 - x^k)^3 * (1 - x^(6*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 09 2017 *)

Formula

a(n) ~ 5^(1/4) * exp(sqrt(10*n)*Pi/3) / (9*2^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 09 2017
Showing 1-3 of 3 results.

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