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A326656 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 colors in (weakly) increasing order. 2
0, 1, 6, 34, 191, 1208, 7840, 54152, 377396, 2868528, 22719712, 187318016, 1594593876, 13795808224, 125535871760, 1192418406800, 11747646588912, 118703814213296, 1223646182128656, 12755728151091424, 137199027931128992, 1527404635450188128, 17599899510211606336 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

FORMULA

a(n) = Sum_{k=1..n} k * A326500(n,k).

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->

      b(n-t, min(n-t, i-1), k)*binomial(k+t-1, 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=0..n):

seq(a(n), n=0..25);

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k + t - 1, t]], {j, 0, n/i}]]];

a[n_] := Sum[k Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];

a /@ Range[0, 25] (* Jean-Fran├žois Alcover, Dec 15 2020, after Alois P. Heinz *)

CROSSREFS

Cf. A326500.

Sequence in context: A052264 A049608 A244937 * A154244 A273583 A126501

Adjacent sequences:  A326653 A326654 A326655 * A326657 A326658 A326659

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 12 2019

STATUS

approved

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Last modified January 21 18:19 EST 2022. Contains 350479 sequences. (Running on oeis4.)