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

0,3

Links

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

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<h(n), 0, add(
      (t-> 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 *)

Crossrefs

Row sums of A326616 and of A326617.

Sequence in context: A150372 A150373 A150374 * A150375 A150376 A150377

Adjacent sequences: A326645 A326646 A326647 * A326649 A326650 A326651

Keyword

nonn

Author

Alois P. Heinz, Sep 12 2019

Status

approved