cp's OEIS Frontend

A326654 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 colors in (weakly) increasing order.

%I A326654 #26 Jun 01 2025 19:57:42
%S A326654 1,1,4,16,70,356,1928,11428,69772,471200,3350320,25067040,195361800,
%T A326654 1559368544,13165162256,116528178688,1074460079840,10203335290992,
%U A326654 99238550358000,979455883492672,10002569256970848,105957081274335392,1164108439659208704
%N A326654 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 colors in (weakly) increasing order.
%H A326654 Alois P. Heinz, <a href="/A326654/b326654.txt">Table of n, a(n) for n = 0..300</a>
%p A326654 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A326654       b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
%p A326654     end:
%p A326654 a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
%p A326654 seq(a(n), n=0..25);
%t A326654 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 A326654 a[n_] := Sum[Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t A326654 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)
%Y A326654 Row sums of A326500.
%K A326654 nonn
%O A326654 0,3
%A A326654 _Alois P. Heinz_, Sep 12 2019