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.
1, 1, 5, 19, 85, 381, 1751, 8135, 38173, 180415, 857695, 4096830, 19645975, 94523729, 456079769, 2206005414, 10693086637, 51930129399, 252617434619, 1230714593340, 6003931991895, 29325290391416, 143393190367102, 701862880794183, 3438561265961263
Offset: 0
Keywords
Examples
a(2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1433
- Wikipedia, Partition (number theory)
Programs
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Maple
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i)))) end: a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n): seq(a(n), n=0..25); # second Maple program: b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add( `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n) end: g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n+1), (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2)))) end: a:= n-> g(n$2): seq(a(n), n=0..25); # Alois P. Heinz, Jan 29 2021 -
Mathematica
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]}]]]; a[n_] := Sum[b[2n, 2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}]; a /@ Range[0, 25] (* Jean-François Alcover, May 06 2020, after Maple *) Table[SeriesCoefficient[(-1 + QPochhammer[-1, Sqrt[x]]/2)^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jan 15 2024 *) (* 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 *)
Formula
a(n) = A308680(2n,n).
a(n) ~ c * d^n / sqrt(n), where d = 5.0032778445310926321307990027... and c = 0.2798596129161126875318997... - Vaclav Kotesovec, Sep 14 2019
a(n) = [x^(2n)] (-1 + Product_{j>=1} (1 + x^j))^n. - Alois P. Heinz, Jan 29 2021