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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.
3
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 12 2019
STATUS
approved