A326616 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), n>=0, A185283(n)<=k<=n, read by rows.

1, 1, 2, 2, 5, 1, 9, 13, 9, 44, 42, 10, 96, 225, 150, 9, 152, 680, 1098, 576, 3, 155, 1350, 4155, 5201, 2266, 124, 2180, 11730, 26642, 26904, 9966, 140, 3751, 30300, 106281, 182000, 149832, 47466, 160, 6050, 69042, 348061, 896392, 1229760, 855240, 237019

Offset

0,3

Comments

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

Links

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

Wikipedia, Partition (number theory)

Formula

Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).

Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).

Example

T(3,2) = 2: 2a1b, 2b1a.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1,  9,  13;
            9,  44,   42;
           10,  96,  225,   150;
            9, 152,  680,  1098,    576;
            3, 155, 1350,  4155,   5201,   2266;
               124, 2180, 11730,  26642,  26904,   9966;
               140, 3751, 30300, 106281, 182000, 149832, 47466;
               ...

Maple

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=h(n)..n), n=0..12);

Mathematica

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]], True, k++,  If[g[k] >= n, Return[k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t,   b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]][i*j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, h[n], n}], {n, 0, 12}]  // Flatten (* Jean-François Alcover, Feb 27 2021, after Alois P. Heinz *)

Crossrefs

Main diagonal gives A178682.

Row sums give A326648.

Column sums give A326650.

Cf. A000203, A185283, A326617 (this triangle read by columns), A326649, A326651.

Sequence in context: A165922 A337293 A307834 * A249033 A068762 A326914

Adjacent sequences: A326613 A326614 A326615 * A326617 A326618 A326619

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 12 2019

Status

approved