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A324595 Number of colored integer partitions of 2n such that all colors from an n-set are used and parts differ by size or by color.

%I A324595 #32 Jan 15 2024 12:34:00
%S A324595 1,1,5,19,85,381,1751,8135,38173,180415,857695,4096830,19645975,
%T A324595 94523729,456079769,2206005414,10693086637,51930129399,252617434619,
%U A324595 1230714593340,6003931991895,29325290391416,143393190367102,701862880794183,3438561265961263
%N A324595 Number of colored integer partitions of 2n such that all colors from an n-set are used and parts differ by size or by color.
%H A324595 Alois P. Heinz, <a href="/A324595/b324595.txt">Table of n, a(n) for n = 0..1433</a>
%H A324595 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A324595 a(n) = A308680(2n,n).
%F A324595 a(n) ~ c * d^n / sqrt(n), where d = 5.0032778445310926321307990027... and c = 0.2798596129161126875318997... - _Vaclav Kotesovec_, Sep 14 2019
%F A324595 a(n) = [x^(2n)] (-1 + Product_{j>=1} (1 + x^j))^n. - _Alois P. Heinz_, Jan 29 2021
%e A324595 a(2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
%p A324595 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A324595       b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
%p A324595     end:
%p A324595 a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p A324595 seq(a(n), n=0..25);
%p A324595 # second Maple program:
%p A324595 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A324595      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A324595     end:
%p A324595 g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n+1),
%p A324595       (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A324595     end:
%p A324595 a:= n-> g(n$2):
%p A324595 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 29 2021
%t A324595 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k] Binomial[k, j]][n - i j], {j, 0, Min[k, n/i]}]]];
%t A324595 a[n_] := Sum[b[2n, 2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
%t A324595 a /@ Range[0, 25] (* _Jean-François Alcover_, May 06 2020, after Maple *)
%t A324595 Table[SeriesCoefficient[(-1 + QPochhammer[-1, Sqrt[x]]/2)^n, {x, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Jan 15 2024 *)
%t A324595 (* Calculation of constant d: *) 1/r /. FindRoot[{2 + 2*s == QPochhammer[-1, Sqrt[r*s]], Sqrt[r]*Derivative[0, 1][QPochhammer][-1, Sqrt[r*s]] == 4*Sqrt[s]}, {r, 1/5}, {s, 1}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Jan 15 2024 *)
%Y A324595 Cf. A000009, A270913, A308680, A340987.
%K A324595 nonn
%O A324595 0,3
%A A324595 _Alois P. Heinz_, Sep 03 2019