A102186 -id:A102186 - cp's OEIS frontend

A077285 Number of partitions of n with designated summands.

1, 1, 3, 5, 10, 15, 28, 41, 69, 102, 160, 231, 352, 498, 732, 1027, 1470, 2031, 2856, 3896, 5382, 7272, 9896, 13233, 17800, 23579, 31362, 41219, 54288, 70791, 92456, 119698, 155097, 199512, 256664, 328134, 419436, 533162, 677412, 856573, 1082284, 1361679

Offset: 0

Jorn B. Olsson (olsson(AT)math.ku.dk), Nov 26 2003

nonn

Sum of products of multiplicities of parts in all partitions of n. The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products are 1,1,2,2,4 and their sum is a(4) = 10. - Vladeta Jovovic, Feb 16 2005

			a(3)=5 because the partitions of 3 with designated summands are 3', 2'1', 1'11, 11'1, 111'.
1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 15*x^5 + 28*x^6 + 41*x^7 + 69*x^8 + 102*x^9 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
G. E. Andrews, R. P. Lewis, and J. Lovejoy, Partitions with designated summands, Acta Arith. 105 (2002), no. 1, 51-66.
William Y. C. Chen, Kathy Q. Ji, Hai-Tao Jin, and Erin Y. Y. Shen, On the Number of Partitions with Designated Summands, arXiv:1208.2210 [math.CO], 2012.
Daniel Herden, Mark R. Sepanski, Jonathan Stanfill, Cordell Hammon, Joel Henningsen, Henry Ickes, and Indalecio Ruiz, Partitions With Designated Summands Not Divisible by 2^L, 2, and 3^L Modulo 2, 4, and 3, arXiv:2101.04058 [math.CO], 2021. See also Integers (2023) Vol. 23, Art. No. A43.
N. J. A. Sloane, Transforms

Cf. A000041, A091601.
Cf. A102186 (partitions into odd parts with designated summands).
Cf. A258210, A266477.

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:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 26 2013

max = 50; f = Product[(1-x^i+x^(2*i))/(1-x^i)^2, {i, 1, max}]; s = Series[f, {x, 0, max}] // Normal; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, May 06 2014, after Vladeta Jovovic *)
nmax=100; CoefficientList[Series[Product[(1+x^(3*k)) / ((1-x^k) * (1-x^(2*k))), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)
QP = QPochhammer; s = QP[q^6]/(QP[q]*QP[q^2]*QP[q^3]) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)
Table[Total[l = Tally /@ IntegerPartitions@n;
Table[x = l[[i]]; Product[x[[j, 2]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 41}] (* Robert Price, Jun 06 2020 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)), n))} /* Michael Somos, Feb 05 2004 */

Expansion of eta(q^6) / (eta(q) * eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Feb 05 2004
Euler transform of period 6 sequence [ 1, 2, 2, 2, 1, 2, ...]. - Michael Somos, Feb 05 2004
G.f.: P(x)*P(x^2)*P(x^3)/P(x^6), where P(x)=Product_{k>0} 1/(1-x^k) is the partition generating function (A000041).
Equals EULER(DCONV(A000012, iEULER(A000027))).
G.f.: Product_{i>=1} (1-x^i+x^(2*i)) / (1-x^i)^2. - Vladeta Jovovic, Jan 16 2005
a(n) ~ 5^(3/4) * exp(Pi*sqrt(10*n)/3) / (2^(11/4) * 3^(3/2) * n^(5/4)). - Vaclav Kotesovec, Nov 28 2015
a(n) = Sum_{k>=1} k*A266477(n,k). - Alois P. Heinz, Dec 29 2015
G.f.: Product_{i>0} (1 + Sum_{j>0} j*x^(j*i)). - Seiichi Manyama, Oct 08 2017

Edited and extended by Christian G. Bower, Jan 23 2004

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

Seiichi Manyama, Oct 08 2017

nonn

			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.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Bernard L. S. Lin, The number of tagged parts over the partitions with designated summands, Journal of Number Theory.

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

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 *)

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)

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

A293461 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(j(2*i-1))).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 4, 1, 1, 0, 1, 1, 2, 4, 1, 3, 1, 0, 1, 1, 2, 4, 5, 3, 3, 1, 0, 1, 1, 2, 4, 5, 3, 6, 5, 2, 0, 1, 1, 2, 4, 5, 8, 6, 5, 6, 2, 0, 1, 1, 2, 4, 5, 8, 6, 9, 9, 4, 2, 0, 1, 1, 2, 4, 5, 8, 12, 9, 9, 13

Offset: 0

Seiichi Manyama, Oct 09 2017

			Square array begins:
   1, 1, 1, 1, 1, ...
   0, 1, 1, 1, 1, ...
   0, 0, 2, 2, 2, ...
   0, 1, 1, 4, 4, ...
   0, 1, 1, 1, 5, ...
   0, 1, 3, 3, 3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000007, A000700, A293304, A293463.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A102186.
Cf. A290216.

max = 12; A[n_, k_] := SeriesCoefficient[Product[(x*(-(k*x^((2*i - 1)*(k + 1) + 1)) - x^((2*i - 1)*(k + 1) + 1) + k*x^((2*i - 1)*(k + 1) + 2*i) + x^(2*i)))/(x^(2*i) - x)^2 + 1, {i, 1, max}], {x, 0, n}]; Flatten[ Table[ A[n - k, k], {n, 0, max}, {k, n, 0, -1}]] (* Jean-François Alcover, Oct 10 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

A293628 Expansion of Product_{k>0} ((1 - q^(2k))^3(1 - q^(6k))(1 - q^(12k)))/((1 - q^k)^4(1 - q^(4*k))).

Original entry on oeis.org

1, 4, 11, 28, 64, 136, 274, 528, 982, 1772, 3115, 5352, 9012, 14904, 24252, 38888, 61527, 96156, 148584, 227204, 344056, 516296, 768206, 1133952, 1661326, 2416816, 3492442, 5014932, 7157996, 10158672, 14339032, 20134888, 28133641, 39124028, 54161282, 74652260

Offset: 0

Seiichi Manyama, Oct 13 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
G. E. Andrews, R. P. Lewis, J. Lovejoy, Partitions with designated summands, Acta Arith. 105 (2002), no. 1, 51-66.

Cf. A293426, A293629.

Cf. A102186 (PDO(n)).

nmax = 50; CoefficientList[Series[Product[(1+x^k)^3 * (1-x^(6*k)) * (1-x^(12*k)) / ((1-x^k) * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 15 2017 *)

a(n) = (1/2) * A102186(3*n+2).

a(n) ~ 5^(1/4) * exp(sqrt(5*n/3)*Pi) / (2^(7/2) * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 15 2017

A329163 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(j(2k - 1))).

Original entry on oeis.org

1, 1, 3, 9, 22, 59, 156, 405, 1061, 2786, 7284, 19071, 49948, 130738, 342288, 896175, 2346134, 6142287, 16080852, 42100020, 110219366, 288558380, 755455128, 1977807393, 5177967900, 13556094631, 35490316938, 92914858431, 243254253904, 636847905903, 1667289469704, 4365020491362

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

nonn

Weigh transform of A032198.

Cf. A032198, A102186, A329156.

nmax = 31; CoefficientList[Series[Product[1/(1 - Sum[j x^(j (2 k - 1)), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Product[1/(1 - x^(2 k - 1)/(1 - x^(2 k - 1))^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1 / (1 - x^(2*k - 1) / (1 - x^(2*k - 1))^2).
G.f.: Product_{k>=1} (1 + x^k)^A032198(k).
a(n) ~ c * phi^(2*n) / sqrt(5), where c = Product_{k>=2} 1/(1 - phi^(2 - 4*k)/(phi^(2 - 4*k) - 1)^2) = 1.07705428718361459418304978675229012... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 07 2019

cp's OEIS Frontend

A077285 Number of partitions of n with designated summands.

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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).

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A293461 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*(2*i-1))).

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

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A293628 Expansion of Product_{k>0} ((1 - q^(2*k))^3*(1 - q^(6*k))*(1 - q^(12*k)))/((1 - q^k)^4*(1 - q^(4*k))).

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A329163 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(j*(2*k - 1))).

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A293461 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(j(2*i-1))).

A293628 Expansion of Product_{k>0} ((1 - q^(2k))^3(1 - q^(6k))(1 - q^(12k)))/((1 - q^k)^4(1 - q^(4*k))).

A329163 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(j(2k - 1))).