A381895 Triangle read by rows: T(n, k) is the number of partitions of n with at most k parts where 0 <= k <= n, and each part is one of two kinds.

1, 0, 2, 0, 2, 5, 0, 2, 6, 10, 0, 2, 9, 15, 20, 0, 2, 10, 22, 30, 36, 0, 2, 13, 31, 48, 58, 65, 0, 2, 14, 40, 68, 90, 102, 110, 0, 2, 17, 51, 97, 135, 162, 176, 185, 0, 2, 18, 64, 128, 194, 242, 274, 290, 300, 0, 2, 21, 77, 171, 271, 357, 415, 452, 470, 481

Offset: 0

Author

Peter Dolland, Mar 09 2025

Keywords

• nonn
• tabl

Comments

The 1-kind case is Euler's table A026820.

Examples

			Triangle starts:
   0 : [1]
   1 : [0, 2]
   2 : [0, 2,  5]
   3 : [0, 2,  6, 10]
   4 : [0, 2,  9, 15,  20]
   5 : [0, 2, 10, 22,  30,  36]
   6 : [0, 2, 13, 31,  48,  58,  65]
   7 : [0, 2, 14, 40,  68,  90, 102, 110]
   8 : [0, 2, 17, 51,  97, 135, 162, 176, 185]
   9 : [0, 2, 18, 64, 128, 194, 242, 274, 290, 300]
  10 : [0, 2, 21, 77, 171, 271, 357, 415, 452, 470, 481]
  ...
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened

Crossrefs

Main diagonal gives A000712.

Cf. A026820.

Programs

PARI

A381895(row_max) = {my(N=row_max+1,x='x+O('x^N), y='y+O('y^N), h=prod(i=1,N, 1/(1-y*x^i)^2)/(1-y)); for(n=0,N-1, if(n<1, print([1]),print(concat([0],Vec(polcoeff(h, n))[1..n]))))}
A381895(12) \\ John Tyler Rascoe, Mar 19 2025

Python

from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
def a381895_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= 1 + p[k]
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

Formula

G.f.: A(x,y,2) where A(x,y,p) = 1/(1-y) * Product_{i>0} 1/(1-y*x^i)^p is the g.f for the number of partitions of n with at most k parts and p kinds of each part. - John Tyler Rascoe, Mar 19 2025