A326654 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order.

1, 1, 4, 16, 70, 356, 1928, 11428, 69772, 471200, 3350320, 25067040, 195361800, 1559368544, 13165162256, 116528178688, 1074460079840, 10203335290992, 99238550358000, 979455883492672, 10002569256970848, 105957081274335392, 1164108439659208704

Offset: 0

Author

Alois P. Heinz, Sep 12 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Row sums of A326500.

Programs

Maple

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-1, t))(i*j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k + t - 1, t]], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)