A327679 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

1, 1, 4, 18, 112, 732, 6156, 53720, 559584, 6138216, 76636080, 1006039320, 14693223032, 224774090592, 3756082129296, 65650522695344, 1236568354232176, 24299076684879264, 509677108276779168, 11124779898457678240, 257204596479739401760, 6174928911548312072704

Offset: 0

Author

Alois P. Heinz, Sep 21 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Row sums of A309973.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
       binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..22);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i-1], k]Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 22] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)