A308680 -id:A308680 - cp's OEIS frontend

A327388 Number of colored integer partitions of n such that ten colors are used and parts differ by size or by color.

1, 10, 65, 320, 1320, 4762, 15500, 46410, 129710, 341990, 857695, 2059430, 4759235, 10630810, 23034880, 48562378, 99866045, 200766810, 395317950, 763661010, 1449390299, 2706189810, 4976391015, 9021860260, 16139848000, 28515535112, 49792637480, 85989053350

Offset: 10

Alois P. Heinz, Sep 03 2019

nonn

In general, column k > 0 of A308680 is asymptotic to exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

Alois P. Heinz, Table of n, a(n) for n = 10..10000 (terms n = 5001..8000 from Vaclav Kotesovec)
Wikipedia, Partition (number theory)

Column k=10 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))(10):
seq(a(n), n=10..45);

A327388[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^10, {x, 0, n}]; Table[A327388[n], {n, 10, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

a(n) ~ exp(Pi*sqrt(10*n/3)) * 5^(1/4) / (2^(25/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^10. - Ilya Gutkovskiy, Jan 31 2021

A327380 Number of colored integer partitions of n such that two colors are used and parts differ by size or by color.

Original entry on oeis.org

1, 2, 5, 8, 14, 22, 34, 50, 73, 104, 146, 202, 275, 372, 498, 660, 868, 1134, 1470, 1896, 2430, 3098, 3931, 4964, 6240, 7814, 9746, 12110, 14997, 18510, 22772, 27934, 34166, 41672, 50698, 61520, 74470, 89940, 108378, 130312, 156364, 187244, 223785, 266962

Offset: 2

Alois P. Heinz, Sep 03 2019

nonn

With offset 0 convolution square of A000009(k+1). - George Beck, Jan 28 2021

			a(4) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
a(5) = 8: 2a2b1a, 2a2b1b, 3a1a1b, 3b1a1b, 3a2b, 3b2a, 4a1b, 4b1a.

Vaclav Kotesovec, Table of n, a(n) for n = 2..10000 (terms 2..5000 from Alois P. Heinz)
Wikipedia, Partition (number theory)

Column k=2 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))(2):
seq(a(n), n=2..45);

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_] := With[{k = 2}, Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]];
a /@ Range[2, 45] (* Jean-François Alcover, May 06 2020, after Maple *)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 14 2019

G.f.: (-1 + Product_{j>=1} (1 + x^j))^2. - Alois P. Heinz, Jan 29 2021

A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 3, 3, 4, 0, 1, 0, 1, 4, 6, 4, 5, 0, 1, 0, 2, 5, 9, 10, 5, 6, 0, 1, 0, 2, 8, 13, 16, 15, 6, 7, 0, 1, 0, 2, 9, 21, 26, 25, 21, 7, 8, 0, 1, 0, 2, 12, 27, 44, 45, 36, 28, 8, 9, 0, 1, 0, 3, 15, 40, 63, 80, 71, 49, 36, 9, 10, 0, 1

Offset: 0

Ilya Gutkovskiy, Feb 08 2021

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  0,  1;
  0,  1,  0,  1;
  0,  1,  2,  0,  1;
  0,  1,  2,  3,  0,  1;
  0,  1,  3,  3,  4,  0,  1;
  0,  1,  4,  6,  4,  5,  0,  1;
  0,  2,  5,  9, 10,  5,  6,  0,  1;
  0,  2,  8, 13, 16, 15,  6,  7,  0,  1;
  0,  2,  9, 21, 26, 25, 21,  7,  8,  0,  1;
  0,  2, 12, 27, 44, 45, 36, 28,  8,  9,  0,  1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give A000007, A000700, A338463, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253.
Main diagonal and lower diagonals give A000012, A000004, A001477, A000217, A000290.
Row sums give A307058.
T(2n,n) gives A341265.
Cf. A000009, A047265, A060642, A286352, A308680.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
T:= proc(n, k) option remember;
      `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 09 2021

T[n_, k_] := SeriesCoefficient[(-1 + 2/QPochhammer[-1, -x])^k, {x, 0, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

G.f. of column k: (-1 + Product_{j>=1} (1 + x^(2*j-1)))^k.

Sum_{k=0..n} (-1)^(n-k) * T(n,k) = A000009(n).

A325915 Total number of colors used in all colored integer partitions of n where all colors from an initial interval of the color palette are used and parts differ by size or by color.

Original entry on oeis.org

0, 1, 3, 9, 25, 67, 176, 453, 1149, 2882, 7161, 17654, 43238, 105303, 255210, 615896, 1480771, 3548313, 8477415, 20199596, 48014369, 113879450, 269555798, 636875077, 1502195104, 3537705916, 8319377813, 19537936874, 45827441193, 107366261405, 251268532266

Offset: 0

Alois P. Heinz, Sep 08 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2807

Cf. A000009, A304969, A308680.

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) option remember; `if`(n=0, [1, 0],
      (p-> p+[0, p[1]])(add(b(j)*g(n-j), j=1..n)))
    end:
a:= n-> g(n)[2]:
seq(a(n), n=0..32);

b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j] Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
g[n_] := g[n] = If[n == 0, {1, 0}, Function[p, p + {0, p[[1]]}][Sum[b[j] g[n - j], {j, 1, n}]]];
a[n_] := g[n][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A308680(n,k).

a(n) ~ c * d^n * n, where d = 2.26562663992642295791262530033324290454663... is the root of the equation QPochhammer[-1, 1/d] = 4 and c = 0.1771510533646387556482103930322780317974659818141571819... - Vaclav Kotesovec, Sep 18 2019

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A327388 Number of colored integer partitions of n such that ten colors are used and parts differ by size or by color.

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A327380 Number of colored integer partitions of n such that two colors are used and parts differ by size or by color.

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A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

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A325915 Total number of colors used in all colored integer partitions of n where all colors from an initial interval of the color palette are used and parts differ by size or by color.

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Maple

Mathematica

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