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A351726
Table T(n,k) read by rows: number of compositions of n into k parts of size 1, 5, 10 or 25.
2
1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 6, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 3, 6, 0, 0, 0, 8, 0, 0, 0, 1, 0, 0, 0, 0, 4
OFFSET
0,24
FORMULA
T(n,0) = 0 if k>0.
G.f.: 1/(1-y*g(x)) where g(x)=x+x^5+x^10+x^25 is the g.f. of column k=1.
EXAMPLE
T(7,3)=3 counts 1+1+5 =1+5+1 =5+1+1.
T(10,2)=1 counts 5+5.
T(12,3)=3 counts 1+1+10 =1+10+1 =10+1+1.
T(15,3)=1 counts 5+5+5.
T(16,3)=6 counts 1+5+10 =1+10+5 =5+1+10 =5+10+1 =10+1+5 =10+5+1.
The triangle starts in row n=0 and columns 0<=k<=n:
1
0 1
0 0 1
0 0 0 1
0 0 0 0 1
0 1 0 0 0 1
0 0 2 0 0 0 1
0 0 0 3 0 0 0 1
0 0 0 0 4 0 0 0 1
0 0 0 0 0 5 0 0 0 1
0 1 1 0 0 0 6 0 0 0 1
0 0 2 3 0 0 0 7 0 0 0 1
0 0 0 3 6 0 0 0 8 0 0 0 1
0 0 0 0 4 10 0 0 0 9 0 0 0 1
0 0 0 0 0 5 15 0 0 0 10 0 0 0 1
0 0 2 1 0 0 6 21 0 0 0 11 0 0 0 1
0 0 0 6 4 0 0 7 28 0 0 0 12 0 0 0 1
0 0 0 0 12 10 0 0 8 36 0 0 0 13 0 0 0 1
0 0 0 0 0 20 20 0 0 9 45 0 0 0 14 0 0 0 1
0 0 0 0 0 0 30 35 0 0 10 55 0 0 0 15 0 0 0 1
0 0 1 3 1 0 0 42 56 0 0 11 66 0 0 0 16 0 0 0 1
0 0 0 3 12 5 0 0 56 84 0 0 12 78 0 0 0 17 0 0 0 1
0 0 0 0 6 30 15 0 0 72 120 0 0 13 91 0 0 0 18 0 0 0 1
CROSSREFS
Cf. A351724 (row sums), A351725 (partitions).
Sequence in context: A343221 A327169 A299173 * A285720 A269175 A086076
KEYWORD
nonn,easy,tabl
AUTHOR
R. J. Mathar, Feb 17 2022
STATUS
approved