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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1433

Wikipedia, Partition (number theory)

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

Example

a(2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.

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 *)

Crossrefs

Cf. A000009, A270913, A308680, A340987.

Sequence in context: A149796 A348410 A005191 * A275027 A364743 A147091

Adjacent sequences: A324592 A324593 A324594 * A324596 A324597 A324598

Keyword

nonn

Author

Alois P. Heinz, Sep 03 2019

Status

approved