A351726 Table T(n,k) read by rows: number of compositions of n into k parts of size 1, 5, 10 or 25.

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

Author

R. J. Mathar, Feb 17 2022

Keywords

• nonn
• easy
• tabl

Examples

			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
		

Links

• L. Zhiwen, Expect coin value problem (a variant of coin exchange problem), math.stackexchange Feb 12, 2022.
• Index entries for sequences related to making change.

Crossrefs

Cf. A351724 (row sums), A351725 (partitions).

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.