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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

%I

%S 0,1,6,34,191,1208,7840,54152,377396,2868528,22719712,187318016,

%T 1594593876,13795808224,125535871760,1192418406800,11747646588912,

%U 118703814213296,1223646182128656,12755728151091424,137199027931128992,1527404635450188128,17599899510211606336

%N 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.

%H Alois P. Heinz, <a href="/A326656/b326656.txt">Table of n, a(n) for n = 0..300</a>

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

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

%p b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))

%p end:

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

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

%t 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}]]];

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

%t a /@ Range[0, 25] (* _Jean-Fran├žois Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)

%Y Cf. A326500.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 12 2019

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Last modified May 16 04:32 EDT 2022. Contains 353688 sequences. (Running on oeis4.)