A326648 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 distinct colors in increasing order.

1, 1, 2, 7, 23, 95, 481, 2515, 13130, 77546, 519770, 3641724, 25931163, 185418629, 1411248697, 11735504788, 103340890753, 931471895697, 8448978391755, 76541843977198, 715994685630321, 7110500945450780, 74757652968961770, 815423663501064107, 9012653697655462141

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 A326616 and of A326617.

Programs

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 or k b(n-t, min(n-t, i-1), k)*binomial(k, 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=h(n)..n):
seq(a(n), n=0..25);

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 || k < h[n], 0, Sum[With[ {t = i j}, b[n-t, Min[n-t, i-1], k] Binomial[k, t]], {j, 0, n/i}]]];
a[n_] := Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {k, h[n], n}, {i, 0, k}];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)