A077285 -id:A077285 - cp's OEIS frontend

A369704 Number of pairs (p,q) of partitions of n such that the set of parts in q is a subset of the set of parts in p.

1, 1, 2, 4, 8, 13, 28, 43, 84, 137, 243, 372, 684, 1010, 1702, 2620, 4256, 6276, 10134, 14740, 23094, 33742, 51139, 73550, 111303, 158140, 233006, 331099, 481324, 674778, 973928, 1353504, 1925734, 2668263, 3748636, 5153887, 7201684, 9820055, 13572468, 18445878

Offset: 0

Alois P. Heinz, Jan 29 2024

nonn

			a(5) = 13: (11111, 11111), (2111, 11111), (2111, 2111), (2111, 221), (221, 11111), (221, 2111), (221, 221), (311, 11111), (311, 311), (32, 32), (41, 11111), (41, 41), (5, 5).

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A000041, A001255, A077285, A297388, A369707, A369910.

b:= proc(n, m, i) option remember; `if`(n=0,
     `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(
      add(b(n-i*j, m-i*h, i-1), h=0..m/i), j=1..n/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..42);

b[n_, m_, i_] := b[n, m, i] = If[n == 0, If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i - 1] + Sum[Sum[b[n - i*j, m - i*h, i - 1], {h, 0, m/i}], { j, 1, n/i}]]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

a(n) = A000041(n) + A369707(n).

A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 8, 1, 1, 15, 4, 1, 21, 13, 1, 33, 28, 1, 1, 41, 58, 4, 1, 56, 103, 13, 1, 69, 170, 35, 1, 87, 269, 77, 1, 1, 99, 404, 158, 4, 1, 127, 579, 298, 13, 1, 141, 810, 529, 35, 1, 165, 1116, 880, 86, 1, 189, 1470, 1431, 183, 1, 1, 220, 1935, 2214, 371, 4, 1, 238, 2475, 3348, 701, 13

Offset: 0

Omar E. Pol, Jul 22 2025

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).
Column k gives the partial sums of the column k of A385001.
Columns converge to A210843 which is also the partial sums of A000716.

			Triangle begins:
---------------------------------------------
   n\k:   0    1     2      3     4    5   6
---------------------------------------------
   0 |    1;
   1 |    1,   1;
   2 |    1,   4;
   3 |    1,   8,    1;
   4 |    1,  15,    4;
   5 |    1,  21,   13;
   6 |    1,  33,   28,     1;
   7 |    1,  41,   58,     4;
   8 |    1,  56,  103,    13;
   9 |    1,  69,  170,    35;
  10 |    1,  87,  269,    77,    1;
  11 |    1,  99,  404,   158,    4;
  12 |    1, 127,  579,   298,   13;
  13 |    1, 141,  810,   529,   35;
  14 |    1, 165, 1116,   880,   86;
  15 |    1, 189, 1470,  1431,  183,   1;
  16 |    1, 220, 1935,  2214,  371,   4;
  17 |    1, 238, 2475,  3348,  701,  13;
  18 |    1, 277, 3156,  4894, 1269,  35;
  19 |    1, 297, 3921,  7036, 2187,  86;
  20 |    1, 339, 4866,  9871, 3639, 194;
  21 |    1, 371, 5906, 13629, 5872, 402,  1;
  ...

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

Columns: A000012 (k=0), A024916 (k=1).

Cf. A000217, A000716, A002024, A003056, A077285, A195825, A210843, A211970, A385001.

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:
g:= proc(n) option remember; `if`(n<0, 0, g(n-1)+b(n$2)) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 22 2025

A163318 Expansion of g.f.: Product_{k>=1} 1+k*x^k/(1-x^k)^2.

Original entry on oeis.org

1, 1, 4, 8, 19, 36, 76, 142, 272, 496, 900, 1592, 2784, 4792, 8138, 13688, 22703, 37380, 60838, 98310, 157298, 250162, 394332, 618032, 961512, 1487563, 2286610, 3496776, 5316666, 8044598, 12110538, 18147166, 27068692, 40203306, 59459998, 87587428, 128522850

Offset: 0

Vladeta Jovovic, Jul 24 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006906, A077285, A162506.

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1) +add(b(n-i*j, i-1)*(j*i), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 25 2013

terms = 40;
CoefficientList[Product[1 + k x^k/(1 - x^k)^2, {k, 1, terms}] + O[x]^terms, x] (* Jean-François Alcover, Nov 12 2020 *)

A293378 Expansion of (eta(q^6)/(eta(q)eta(q^2)eta(q^3)))^2 in powers of q.

Original entry on oeis.org

1, 2, 7, 16, 39, 80, 171, 328, 638, 1168, 2133, 3744, 6540, 11092, 18687, 30816, 50421, 81136, 129582, 204160, 319340, 493952, 758781, 1154624, 1745748, 2617958, 3902614, 5776144, 8501784, 12434320, 18092565, 26175784, 37689734, 53989056, 76993497, 109284736

Offset: 0

Seiichi Manyama, Oct 07 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A077285, A293377.

nmax = 50; CoefficientList[Series[Product[((1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 11 2017 *)

G.f.: Product_{k>0} ((1 - x^(6*k))/((1 - x^k)*(1 - x^(2*k))*(1 - x^(3*k))))^2.

a(n) ~ 5^(5/4) * exp(2*Pi*sqrt(5*n)/3) / (72 * sqrt(3) * n^(7/4)). - Vaclav Kotesovec, Oct 11 2017

A293569 Partitions with designated summands in which no parts are multiples of 3.

Original entry on oeis.org

1, 1, 3, 4, 9, 12, 21, 29, 48, 64, 99, 132, 195, 257, 366, 480, 666, 864, 1173, 1511, 2016, 2576, 3384, 4296, 5574, 7027, 9015, 11296, 14355, 17880, 22527, 27908, 34896, 43008, 53406, 65508, 80844, 98711, 121128, 147272, 179784, 217704, 264489, 319064

Offset: 0

Seiichi Manyama, Oct 12 2017

nonn

			n = 3        n = 4            n = 5
----------   --------------   ------------------
2'+ 1'       4'               5'
1'+ 1 + 1    2'+ 2            4'+ 1'
1 + 1'+ 1    2 + 2'           2'+ 2 + 1'
1 + 1 + 1'   2'+ 1'+ 1        2 + 2'+ 1'
             2'+ 1 + 1'       2'+ 1'+ 1 + 1
             1'+ 1 + 1 + 1    2'+ 1 + 1'+ 1
             1 + 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'
----------   --------------   ------------------
a(3) = 4.    a(4) = 9.        a(5) = 12.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A077285 (PD(n)), A102186 (PDO(n)), A293629.

nmax = 50; CoefficientList[Series[Product[(1-x^(6*k))^2 / ( (1-x^k)^2 * (1+x^k) * (1+x^(9*k)) ), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2017 *)

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(k, n)
  partition(n, 1, n).select{|i| i.all?{|j| j % k > 0}}.map{|a| a.each_with_object(Hash.new(0)){|v, o| o[v] += 1}.values.inject(:*)}.inject(:+)
end
def A293569(n)
  [1] + (1..n).map{|i| A(3, i)}
end
p A293569(40)

Expansion of eta(q^6)^2 * eta(q^9) / (eta(q) * eta(q^2) * eta(q^18)) in powers of q.

a(n) ~ 5^(1/4) * exp(2*Pi*sqrt(5*n/3)/3) / (2 * 3^(7/4)* n^(3/4)). - Vaclav Kotesovec, Oct 13 2017

A384998 Total number of partitions of all numbers <= n with designated summands, n >= 0.

Original entry on oeis.org

1, 2, 5, 10, 20, 35, 63, 104, 173, 275, 435, 666, 1018, 1516, 2248, 3275, 4745, 6776, 9632, 13528, 18910, 26182, 36078, 49311, 67111, 90690, 122052, 163271, 217559, 288350, 380806, 500504, 655601, 855113, 1111777, 1439911, 1859347, 2392509, 3069921, 3926494

Offset: 0

Omar E. Pol, Aug 06 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Partial sums of A077285.
Row sums of A384999.
Cf. A385001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*j, j=1..n/i)))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
seq(a(n), n=0..41);  # Alois P. Heinz, Aug 06 2025

nmax = 50; CoefficientList[Series[1/(1-x) * Product[(1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 08 2025 *)

From Vaclav Kotesovec, Aug 08 2025: (Start)
a(n) ~ 5^(1/4) * exp(sqrt(10*n)*Pi/3) / (2^(9/4) * sqrt(3) * Pi * n^(3/4)).
G.f.: 1/(1-x) * Product_{k>=1} (1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))). (End)

A091601 Number of compositions (ordered partitions) of n with designated summands.

Original entry on oeis.org

1, 1, 3, 6, 14, 30, 69, 153, 345, 771, 1730, 3873, 8682, 19450, 43590, 97668, 218864, 490416, 1098933, 2462458, 5517870, 12364356, 27705944, 62083134, 139115247, 311727845, 698516370, 1565227653, 3507344882, 7859219406, 17610851898

Offset: 0

Christian G. Bower, Jan 23 2004

nonn

			a(3)=6 because the compositions of 3 with designated summands are
3', 2'1', 1'2', 1'11, 11'1, 111'.
The composition 1121 corresponds to 1'12'1' and 11'2'1'.

N. J. A. Sloane, Transforms

Cf. A077285.

Table[l = Split /@ Flatten[Permutations /@ IntegerPartitions@n, 1];
Total[Table[x = l[[i]]; Product[Length@x[[j]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 15}] (* Robert Price, Jun 07 2020 *)

INVERT(DCONV(A000012, iINVERT(A000027)))

G.f.: 1/(1 - sum(k>0, x^k/(1-x^k+x^(2*k)))). - Vladeta Jovovic, Dec 04 2004

A104510 G.f.: Product_{i>=1} (1 - 2*(-x)^i)/(1 - (-x)^i)^2.

Original entry on oeis.org

0, -1, 2, -4, 4, -7, 4, -5, 0, 5, -18, 23, -46, 65, -82, 108, -132, 152, -164, 168, -144, 132, -48, -39, 212, -365, 658, -947, 1382, -1800, 2394, -2947, 3644, -4289, 5102, -5687, 6392, -6820, 7112, -7139, 6776, -5836, 4338, -2036, -1342, 5585, -11392, 18513, -27456, 37876, -51072, 65488, -82982, 101898

Offset: 1

Vladeta Jovovic, Apr 19 2005

Cf. A000712, A077285.

Cf. A104575.

gf:=product((1-2*(-x)^i)/(1-(-x)^i)^2, i=1..100): s:=series(gf, x, 100): for n from 1 to 99 do printf(`%d,`,coeff(s, x, n)) od: # James Sellers, Apr 22 2005

a(n) = Sum (k(1)-1)*(k(2)-1)*...*(k(n)-1), where the sum is taken over all (k(1), k(2), ..., k(n)) such that k(1) + 2*k(2) + ... + n*k(n) = n, k(i) >= 0, i=1..n.

G.f.: Product_{i>=1} (1 - (-x)^i)^A052823(i). - James Sellers, Apr 22 2005

More terms from James Sellers, Apr 22 2005

A255180 Number of partitions of n in which two summands (of each size) are designated.

Original entry on oeis.org

1, 0, 1, 3, 7, 10, 20, 24, 45, 61, 103, 140, 246, 325, 517, 728, 1086, 1472, 2184, 2918, 4197, 5638, 7875, 10497, 14625, 19272, 26354, 34804, 46992, 61490, 82471, 107163, 142128, 184141, 241701, 311282, 406164, 519755, 672726, 858110, 1102872

Offset: 0

Geoffrey Critzer, Mar 19 2015

nonn

			a(4)=7. In order to designate two summands of each size, the multiplicity of each summand must be at least two. For n=4 we consider only the partitions 2+2 and 1+1+1+1.  In the first case there is 1 way and in the second case there are 6 ways.  1 + 6 = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A077285, A070933 (where any number of summands of any size are designated).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*binomial(j, 2), j=2..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 19 2015

nn = 40; CoefficientList[Series[Product[1 + x^(2 n)/(1 - x^n)^3, {n, 1, nn}], {x, 0, nn}], x]

G.f.: Product_{n>=1} 1 + x^(2*n)/(1 - x^n)^3.

cp's OEIS Frontend

A369704 Number of pairs (p,q) of partitions of n such that the set of parts in q is a subset of the set of parts in p.

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A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).

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A163318 Expansion of g.f.: Product_{k>=1} 1+k*x^k/(1-x^k)^2.

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A293378 Expansion of (eta(q^6)/(eta(q)*eta(q^2)*eta(q^3)))^2 in powers of q.

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A293569 Partitions with designated summands in which no parts are multiples of 3.

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A384998 Total number of partitions of all numbers <= n with designated summands, n >= 0.

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A091601 Number of compositions (ordered partitions) of n with designated summands.

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A104510 G.f.: Product_{i>=1} (1 - 2*(-x)^i)/(1 - (-x)^i)^2.

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A293378 Expansion of (eta(q^6)/(eta(q)eta(q^2)eta(q^3)))^2 in powers of q.