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A327557 Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order. 2
0, 1, 5, 29, 173, 1129, 7933, 59757, 480389, 4102233, 37059485, 352891285, 3530465753, 37001007337, 405191214949, 4625525704837, 54929552638957, 677283511701937, 8655757492783861, 114479050583748677, 1564613481125976373, 22068492671782019793 (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 * A327116(n,k).

MAPLE

C:= binomial:

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

      b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))

    end:

a:= n-> add(k*add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):

seq(a(n), n=0..23);

MATHEMATICA

c = Binomial;

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

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

a /@ Range[0, 23] (* Jean-Fran├žois Alcover, Dec 16 2020, after Alois P. Heinz *)

CROSSREFS

Cf. A327116.

Sequence in context: A088349 A272802 A083066 * A163611 A160906 A163073

Adjacent sequences:  A327554 A327555 A327556 * A327558 A327559 A327560

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 16 2019

STATUS

approved

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Last modified January 23 03:15 EST 2022. Contains 350504 sequences. (Running on oeis4.)