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A326649 Total number of colors in all 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
0, 1, 4, 19, 81, 413, 2439, 14655, 86844, 573196, 4224230, 32280154, 249433713, 1925416359, 15732592327, 139542345546, 1304524118159, 12445507282579, 119198874300879, 1137647406084952, 11183828252431175, 116368970786569604, 1278400213028604214 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=A185283(n)..n} k * A326616(n,k) = Sum_{k=A185283(n)..n} k * A326617(n,k).
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(k*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[k Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, h[n], n}];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A215037 A181300 A027240 * A050914 A017961 A180146
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 12 2019
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)