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

0,3

Links

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

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 *)

Crossrefs

Row sums of A309973.

Sequence in context: A060223 A144085 A003708 * A330353 A000986 A364623

Adjacent sequences: A327676 A327677 A327678 * A327680 A327681 A327682

Keyword

nonn

Author

Alois P. Heinz, Sep 21 2019

Status

approved