A327412
a(n) = multinomial(3*n+2; 2, 3, 3, ..., 3) (n times '3').
Original entry on oeis.org
1, 10, 560, 92400, 33633600, 22870848000, 26072766720000, 46174869861120000, 120054661638912000000, 438679733628584448000000, 2175851478797778862080000000, 14240947928731462652313600000000, 120136636726778618934917529600000000, 1280656547507460077846220865536000000000
Offset: 0
-
a:= n-> combinat[multinomial](3*n+2, 3$n, 2):
seq(a(n), n=0..17); # Alois P. Heinz, Sep 07 2019
-
multinomial[n_, k_List] := n!/Times @@ (k!);
a[n_] := multinomial[3n+2, Join[{2}, Table[3, {n}]]];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 19 2025 *)
-
def a(n): return multinomial([2] + [3] * n)
[a(n) for n in range(15)]
A327410
Numbers represented by the partition coefficients of prime partitions.
Original entry on oeis.org
1, 6, 10, 20, 21, 36, 56, 78, 90, 105, 120, 171, 210, 252, 300, 364, 465, 528, 560, 741, 756, 792, 903, 990, 1140, 1176, 1485, 1540, 1680, 1830, 1953, 1980, 2346, 2520, 2600, 2628, 2775, 3240, 3432, 3570, 4095, 4368, 4851, 4960, 5253, 5460, 5886, 5984, 6105
Offset: 1
(2*n)!/2^n (for n >= 1) is a subsequence because [2,2,...,2] (n times '2') is a prime partition. Similarly A327411(n) is a subsequence because [3,2,2,...,2] (n times '2') is a prime partition. (3*n)!/(6^n) and A327412 are subsequences for the same reason.
The representations are not unique. 1 is the represented by all partitions of the form [p], p prime. For example 210 is represented by [3, 2, 2] and by [19, 2]. The list below shows the partitions with the smallest sum.
1 <- [2],
6 <- [2, 2],
10 <- [3, 2],
20 <- [3, 3],
21 <- [5, 2],
36 <- [7, 2],
56 <- [5, 3],
78 <- [11, 2],
90 <- [2, 2, 2],
105 <- [13, 2],
120 <- [7, 3],
171 <- [17, 2],
210 <- [3, 2, 2],
252 <- [5, 5],
300 <- [23, 2].
Showing 1-2 of 2 results.
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