A075197 -id:A075197 - cp's OEIS frontend

A130273 Refines A075197(n): number of partitions of n balls of n colors. The refinement has shape A000041(n).

1, 4, 2, 9, 24, 5, 20, 84, 54, 132, 15, 35, 240, 320, 630, 780, 720, 52, 66, 570, 870, 2280, 465, 6240, 4440, 1320, 8280, 4050, 203, 105, 1260, 1974, 6720, 2394, 20580, 19740, 14385, 11445, 83160, 31080, 34860, 77910, 23772, 877, 176, 2520, 4312, 17640, 5432

Offset: 1

Alford Arnold, May 19 2007

nonn

a(n) can be calculated by resorting A035206 into Mathematica order vice AS1 ordering and then multiplying term by term with A096443(n).

			The array begins
1
4 2
9 24 5
20 84 54 132 15
...
Row three is (9,24,5) because there are (3, 4,5) cases; and we have (3, 6,1) ways to pick 1,2 or 3 colors.

Cf. A000041, A000110, A035206, A075197, A096443, A114994.

A075196 Table T(n,k) by antidiagonals: T(n,k) = number of partitions of n balls of k colors.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 14, 5, 5, 20, 38, 33, 7, 6, 30, 80, 117, 70, 11, 7, 42, 145, 305, 330, 149, 15, 8, 56, 238, 660, 1072, 906, 298, 22, 9, 72, 364, 1260, 2777, 3622, 2367, 591, 30, 10, 90, 528, 2198, 6174, 11160, 11676, 6027, 1132, 42, 11, 110, 735, 3582, 12292, 28784, 42805, 36450, 14873, 2139, 56

Offset: 1

Christian G. Bower, Sep 07 2002

For k>=1, n->infinity is log(T(n,k)) ~ (1+1/k) * k^(1/(k+1)) * Zeta(k+1)^(1/(k+1)) * n^(k/(k+1)). - Vaclav Kotesovec, Mar 08 2015

			Square array T(n,k) begins:
  1,  2,   3,    4,    5, ...
  2,  6,  12,   20,   30, ...
  3, 14,  38,   80,  145, ...
  5, 33, 117,  305,  660, ...
  7, 70, 330, 1072, 2777, ...

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-10: A000041, A005380, A217093, A255050, A255052, A270239, A270240, A270241, A270242, A270243.
Rows 1-3: A000027, A002378, A162147.
Main diagonal: A075197.
Cf. A255903.

with(numtheory):
A:= proc(n, k) option remember; local d, j;
      `if`(n=0, 1, add(add(d*binomial(d+k-1, k-1),
       d=divisors(j)) *A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 26 2012

Transpose[Table[nn=6;p=Product[1/(1- x^i)^Binomial[i+n,n],{i,1,nn}];Drop[CoefficientList[Series[p,{x,0,nn}],x],1],{n,0,nn}]]//Grid  (* Geoffrey Critzer, Sep 27 2012 *)

T(n,k) = Sum_{i=0..k} C(k,i) * A255903(n,i). - Alois P. Heinz, Mar 10 2015

A344098 a(n) = [x^n] Product_{k>=1} (1 + x^k)^binomial(k+n-1,n-1).

Original entry on oeis.org

1, 1, 4, 29, 221, 2027, 21022, 242209, 3060262, 41936745, 618154670, 9735013136, 162892047930, 2882449728121, 53727527279464, 1051276401060921, 21529017626095851, 460231878244308738, 10246160509840187387, 237067632496414877363, 5689786581042000827057, 141415234722601777758232

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Cf. A075197, A209668, A219555, A305206, A338645, A343200, A344097.

Table[SeriesCoefficient[Product[(1 + x^k)^Binomial[k + n - 1, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 21}]
A[n_, k_] := A[n, k] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(j/d + 1) d Binomial[d + k - 1, k - 1], {d, Divisors[j]}] A[n - j, k], {j, 1, n}]]; a[n_] := A[n, n]; Table[a[n], {n, 0, 21}]

cp's OEIS Frontend

A130273 Refines A075197(n): number of partitions of n balls of n colors. The refinement has shape A000041(n).

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A075196 Table T(n,k) by antidiagonals: T(n,k) = number of partitions of n balls of k colors.

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A344098 a(n) = [x^n] Product_{k>=1} (1 + x^k)^binomial(k+n-1,n-1).

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Mathematica