OFFSET
0,4
COMMENTS
A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).
EXAMPLE
The a(0) = 1 through a(5) = 7 ways:
. 1*1 1*2 1*3 1*4 1*5
0*2+3*1 0*3+4*1 0*4+5*1
1*1+1*2 1*1+1*3 1*1+1*4
1*2+1*1 1*3+1*1 1*2+1*3
3*1+0*2 4*1+0*3 1*3+1*2
1*4+1*1
5*1+0*4
MATHEMATICA
combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n, ptn], {ptn, Join@@Permutations/@Select[IntegerPartitions[n], UnsameQ@@#&]}]], {n, 0, 5}]
PROG
(Python)
from math import factorial
from sympy.utilities.iterables import partitions
def A364909(n):
if n == 0: return 1
aset = tuple(set(p) for p in partitions(n) if max(p.values(), default=0)==1)
return sum(factorial(len(t)) for p in partitions(n) for t in aset if set(p).issubset(t)) # Chai Wah Wu, Sep 21 2023
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 18 2023
EXTENSIONS
a(18)-a(37) from Chai Wah Wu, Sep 21 2023
STATUS
approved