A308680 -id:A308680 - cp's OEIS frontend

A304969 Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

1, 1, 2, 5, 11, 25, 57, 129, 292, 662, 1500, 3398, 7699, 17443, 39519, 89536, 202855, 459593, 1041267, 2359122, 5344889, 12109524, 27435660, 62158961, 140828999, 319065932, 722884274, 1637785870, 3710611298, 8406859805, 19046805534, 43152950024, 97768473163

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A000009.
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{2},{1,1}}    {{1},{2,2,2}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{1},{2},{1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{2},{3},{1,1}}
                                   {{1},{2},{3},{4}}
The non-strict version is A055887.
The strongly normal non-strict version is A063834.
The strongly normal version is A270995.
(End)

			From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose a strict integer partition of each part of an integer composition of n. The a(1) = 1 through a(4) = 11 choices are:
  ((1))  ((2))     ((3))        ((4))
         ((1)(1))  ((21))       ((31))
                   ((1)(2))     ((1)(3))
                   ((2)(1))     ((2)(2))
                   ((1)(1)(1))  ((3)(1))
                                ((1)(21))
                                ((21)(1))
                                ((1)(1)(2))
                                ((1)(2)(1))
                                ((2)(1)(1))
                                ((1)(1)(1)(1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2816
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Partition Function Q
Index entries for sequences related to partitions
Index entries for sequences related to compositions

Cf. A050342, A055887, A067687, A081362, A279785, A299106.
Row sums of A308680.
The unordered version is A089259, non-strict A001970 (row-sums of A061260).
For partitions instead of compositions we have A270995, non-strict A063834.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by sum and length.
Cf. A279784.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

G.f.: 1/(1 - Sum_{k>=1} A000009(k)*x^k).
G.f.: 1/(2 - Product_{k>=1} (1 + x^k)).
G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^(2*k-1))).
G.f.: 1/(2 - exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)))).
a(n) ~ c / r^n, where r = 0.441378990861652015438479635503868737167721352874... is the root of the equation QPochhammer[-1, r] = 4 and c = 0.4208931614610039677452560636348863586180784719323982664940444607322... - Vaclav Kotesovec, May 23 2018

A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 0

Offset: 0

Ilya Gutkovskiy, May 07 2017

A(n,k) is the number of partitions of n into distinct parts (or odd parts) with k types of each part.

			A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).
Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).
Square array begins:
  1,  1,  1,   1,   1,   1,  ...
  0,  1,  2,   3,   4,   5,  ...
  0,  1,  3,   6,  10,  15,  ...
  0,  2,  6,  13,  24,  40,  ...
  0,  2,  9,  24,  51,  95,  ...
  0,  3, 14,  42, 100, 206,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
N. J. A. Sloane, Transforms

Columns k=0-32 give: A000007, A000009, A022567-A022596.
Rows n=0-2 give: A000012, A001477, A000217.
Main diagonal gives A270913.
Antidiagonal sums give A299106.
Cf. A144064, A286352, A308680.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 29 2019

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A308680(n,k-i). - Alois P. Heinz, Aug 29 2019

A324595 Number of colored integer partitions of 2n such that all colors from an n-set are used and parts differ by size or by color.

Original entry on oeis.org

1, 1, 5, 19, 85, 381, 1751, 8135, 38173, 180415, 857695, 4096830, 19645975, 94523729, 456079769, 2206005414, 10693086637, 51930129399, 252617434619, 1230714593340, 6003931991895, 29325290391416, 143393190367102, 701862880794183, 3438561265961263

Offset: 0

Alois P. Heinz, Sep 03 2019

nonn

			a(2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.

Alois P. Heinz, Table of n, a(n) for n = 0..1433
Wikipedia, Partition (number theory)

Cf. A000009, A270913, A308680, A340987.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..25);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n+1),
      (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 29 2021

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k] Binomial[k, j]][n - i j], {j, 0, Min[k, n/i]}]]];
a[n_] := Sum[b[2n, 2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, May 06 2020, after Maple *)
Table[SeriesCoefficient[(-1 + QPochhammer[-1, Sqrt[x]]/2)^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jan 15 2024 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{2 + 2*s == QPochhammer[-1, Sqrt[r*s]], Sqrt[r]*Derivative[0, 1][QPochhammer][-1, Sqrt[r*s]] == 4*Sqrt[s]}, {r, 1/5}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jan 15 2024 *)

a(n) = A308680(2n,n).
a(n) ~ c * d^n / sqrt(n), where d = 5.0032778445310926321307990027... and c = 0.2798596129161126875318997... - Vaclav Kotesovec, Sep 14 2019
a(n) = [x^(2n)] (-1 + Product_{j>=1} (1 + x^j))^n. - Alois P. Heinz, Jan 29 2021

A327381 Number of colored integer partitions of n such that three colors are used and parts differ by size or by color.

Original entry on oeis.org

1, 3, 9, 19, 39, 72, 128, 216, 354, 563, 876, 1335, 1998, 2946, 4284, 6154, 8742, 12294, 17129, 23667, 32442, 44151, 59682, 80169, 107054, 142167, 187812, 246895, 323058, 420852, 545958, 705438, 908043, 1164609, 1488504, 1896174, 2407836, 3048255, 3847716

Offset: 3

Alois P. Heinz, Sep 03 2019

nonn

With offset 0 convolution cube of A000009(k+1). - George Beck, Jan 29 2021

Vaclav Kotesovec, Table of n, a(n) for n = 3..10000 (terms 3..5000 from Alois P. Heinz)
Wikipedia, Partition (number theory)

Column k=3 of A308680.

Cf. A000009.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(3):
seq(a(n), n=3..45);

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k] Binomial[k, j]], {j, 0, Min[k, n/i]}]]];
a[n_] := With[{k = 3}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[3, 45] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

a(n) ~ exp(Pi*sqrt(n)) / (8 * n^(3/4)). - Vaclav Kotesovec, Sep 14 2019

G.f.: (-1 + Product_{m >= 1} (1 + x^m))^3. - George Beck, Jan 29 2021

A327382 Number of colored integer partitions of n such that four colors are used and parts differ by size or by color.

Original entry on oeis.org

1, 4, 14, 36, 85, 180, 360, 680, 1234, 2160, 3674, 6092, 9882, 15724, 24594, 37884, 57553, 86344, 128060, 187948, 273178, 393516, 562158, 796860, 1121375, 1567336, 2176664, 3004692, 4124130, 5630160, 7646916, 10335696, 13905376, 18625564, 24843142, 33003072

Offset: 4

Alois P. Heinz, Sep 03 2019

nonn

With offset 0 four-fold convolution of A000009(k+1). - George Beck, Jan 29 2021

Vaclav Kotesovec, Table of n, a(n) for n = 4..10000 (terms 4..5000 from Alois P. Heinz)
Wikipedia, Partition (number theory)

Column k=4 of A308680.

Cf. A000009.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(4):
seq(a(n), n=4..45);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k] Binomial[k, j]], {j, 0, Min[k, n/i]}]]];
a[n_] := With[{k = 4}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[4, 45] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 14 2019

G.f.: (-1 + Product_{m >= 1} (1 + x^m))^4. - George Beck, Jan 29 2021

A327383 Number of colored integer partitions of n such that five colors are used and parts differ by size or by color.

Original entry on oeis.org

1, 5, 20, 60, 160, 381, 845, 1760, 3495, 6660, 12267, 21935, 38230, 65140, 108785, 178437, 287975, 457965, 718575, 1113680, 1706533, 2587655, 3885615, 5781830, 8530625, 12486429, 18140360, 26169335, 37501595, 53403915, 75597130, 106408670, 148973260, 207496090

Offset: 5

Alois P. Heinz, Sep 03 2019

nonn

With offset 0 five-fold convolution of A000009(k+1). - George Beck, Jan 29 2021

Vaclav Kotesovec, Table of n, a(n) for n = 5..10000 (terms 5..5000 from Alois P. Heinz)
Wikipedia, Partition (number theory)

Column k=5 of A308680.

Cf. A000009.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(5):
seq(a(n), n=5..45);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k] Binomial[k, j]], {j, 0, Min[k, n/i]}]]];
a[n_] := With[{k = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[5, 45] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

a(n) ~ exp(Pi*sqrt(5*n/3)) * 5^(1/4) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{m >= 1} (1 + x^m))^5. - George Beck, Jan 29 2021

A327384 Number of colored integer partitions of n such that six colors are used and parts differ by size or by color.

Original entry on oeis.org

1, 6, 27, 92, 273, 720, 1751, 3978, 8565, 17618, 34878, 66792, 124268, 225384, 399618, 694294, 1184340, 1986900, 3282991, 5349372, 8604978, 13678190, 21503439, 33459222, 51563824, 78751470, 119259576, 179169140, 267154842, 395521482, 581629358, 849846186

Offset: 6

Alois P. Heinz, Sep 03 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 6..10000 (terms 6..5000 from Alois P. Heinz)
Wikipedia, Partition (number theory)

Column k=6 of A308680.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(6):
seq(a(n), n=6..45);

A327384[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^6, {x, 0, n}]; Table[A327384[n], {n, 6, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

a(n) ~ exp(Pi*sqrt(2*n)) / (2^(17/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^6. - Ilya Gutkovskiy, Jan 31 2021

A327385 Number of colored integer partitions of n such that seven colors are used and parts differ by size or by color.

Original entry on oeis.org

1, 7, 35, 133, 434, 1253, 3311, 8135, 18851, 41573, 87920, 179305, 354270, 680631, 1275430, 2337097, 4196717, 7398699, 12826324, 21895160, 36848119, 61201709, 100415175, 162886318, 261422357, 415397836, 653899589, 1020282424, 1578729491, 2423647471, 3693050242

Offset: 7

Alois P. Heinz, Sep 03 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..10000 (terms n = 5001..9000 from Vaclav Kotesovec)
Wikipedia, Partition (number theory)

Column k=7 of A308680.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(7):
seq(a(n), n=7..45);

A327385[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^7, {x, 0, n}]; Table[A327385[n], {n, 7, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

a(n) ~ exp(Pi*sqrt(7*n/3)) * 7^(1/4) / (32 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^7. - Ilya Gutkovskiy, Jan 31 2021

A327386 Number of colored integer partitions of n such that eight colors are used and parts differ by size or by color.

Original entry on oeis.org

1, 8, 44, 184, 654, 2048, 5836, 15400, 38173, 89752, 201740, 436104, 911072, 1846648, 3643360, 7016016, 13217634, 24408992, 44260816, 78923480, 138571450, 239838288, 409619196, 690956800, 1152075322, 1900139104, 3102050748, 5015671600, 8036376650, 12766039888

Offset: 8

Alois P. Heinz, Sep 03 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000
Wikipedia, Partition (number theory)

Column k=8 of A308680.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(8):
seq(a(n), n=8..45);

A327386[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^8, {x, 0, n}]; Table[A327386[n], {n, 8, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(19/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^8. - Ilya Gutkovskiy, Jan 31 2021

A327387 Number of colored integer partitions of n such that nine colors are used and parts differ by size or by color.

Original entry on oeis.org

1, 9, 54, 246, 945, 3186, 9729, 27414, 72315, 180415, 429156, 979425, 2155485, 4593330, 9510624, 19188360, 37815948, 72950634, 138002024, 256405887, 468550278, 843138585, 1495634373, 2617905474, 4525424256, 7731765279, 13065217956, 21849902348, 36184992984

Offset: 9

Alois P. Heinz, Sep 03 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..10000
Wikipedia, Partition (number theory)

Column k=9 of A308680.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(9):
seq(a(n), n=9..45);

A327387[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^9, {x, 0, n}]; Table[A327387[n], {n, 9, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

a(n) ~ exp(Pi*sqrt(3*n)) * 3^(1/4) / (64 * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^9. - Ilya Gutkovskiy, Jan 31 2021

cp's OEIS Frontend

A304969 Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).

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A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.

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A324595 Number of colored integer partitions of 2n such that all colors from an n-set are used and parts differ by size or by color.

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A327381 Number of colored integer partitions of n such that three colors are used and parts differ by size or by color.

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A327382 Number of colored integer partitions of n such that four colors are used and parts differ by size or by color.

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A327383 Number of colored integer partitions of n such that five colors are used and parts differ by size or by color.

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A327384 Number of colored integer partitions of n such that six colors are used and parts differ by size or by color.

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A327385 Number of colored integer partitions of n such that seven colors are used and parts differ by size or by color.