A327383 - OEIS

%I #18 Jan 29 2021 19:41:16

%S 1,5,20,60,160,381,845,1760,3495,6660,12267,21935,38230,65140,108785,

%T 178437,287975,457965,718575,1113680,1706533,2587655,3885615,5781830,

%U 8530625,12486429,18140360,26169335,37501595,53403915,75597130,106408670,148973260,207496090

%N Number of colored integer partitions of n such that five colors are used and parts differ by size or by color.

%C With offset 0 five-fold convolution of A000009(k+1). - _George Beck_, Jan 29 2021

%H Vaclav Kotesovec, <a href="/A327383/b327383.txt">Table of n, a(n) for n = 5..10000</a> (terms 5..5000 from Alois P. Heinz)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%F a(n) ~ exp(Pi*sqrt(5*n/3)) * 5^(1/4) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 16 2019

%F G.f.: (-1 + Product_{m >= 1} (1 + x^m))^5. - _George Beck_, Jan 29 2021

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

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

%p     end:

%p a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(5):

%p seq(a(n), n=5..45);

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

%t a[n_] := With[{k = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];

%t a /@ Range[5, 45] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)

%Y Column k=5 of A308680.

%Y Cf. A000009.

%K nonn

%O 5,2

%A _Alois P. Heinz_, Sep 03 2019