A327412 -id:A327412 - cp's OEIS frontend

A327411 a(n) = multinomial(2*n+3; 3, 2, 2, ..., 2) (n times '2').

1, 10, 210, 7560, 415800, 32432400, 3405402000, 463134672000, 79196028912000, 16631166071520000, 4207685016094560000, 1262305504828368000000, 443069232194757168000000, 179886108271071410208000000, 83647040346048205746720000000, 44165637302713452634268160000000

Offset: 0

Peter Luschny, Sep 07 2019

nonn

Cf. A000457, A327412.

a:= n-> combinat[multinomial](2*n+3, 3, 2$n):
seq(a(n), n=0..17);  # Alois P. Heinz, Sep 07 2019

multinomial[n_, k_List] := n!/Times @@ (k!);
a[n_] := multinomial[2n+3, Join[{3}, Table[2, {n}]]];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 19 2025 *)

def a(n):
    return multinomial([3] + [2] * n)
[a(n) for n in range(20)]

a(n) = 2^(-n-1)*Gamma(2*n + 4)/3.
a(n) = (2*(n-1)^2 + 9*n + 1)*a(n-1)  for n > 0.
a(n) / n! = A000457(n).

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

Peter Luschny, Sep 07 2019

nonn

Given a partition pi = (p1, p2, p3, ...) we call the associated multinomial coefficient (p1+p2+ ...)! / (p1!*p2!*p3! ...) the 'partition coefficient' of pi and denote it by . We say 'k is represented by pi' if k = .

A partition is a prime partition if all parts are prime.

			(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].

George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, arXiv:math/0509470 [math.CO], 2005.
Eric Weisstein's World of Mathematics, Prime Partition

Cf. A000607, A036038, A325306, A000680, A327411, A014606, A327412.

def A327410_list(n):
    res = []
    for k in range(2*n):
        P = Partitions(k, parts_in = prime_range(k+1))
        res += [multinomial(p) for p in P]
    return sorted(Set(res))[:n]
print(A327410_list(20))

cp's OEIS Frontend

A327411 a(n) = multinomial(2*n+3; 3, 2, 2, ..., 2) (n times '2').

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula