A351725 Table T(n,k) read by rows: number of partitions 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, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 0

Author

R. J. Mathar, Feb 17 2022

Keywords

• nonn
• easy
• look
• tabl

Comments

Multiset transform of the binary sequence b(n)=1,1,0,0,0,1,0,0,0,0,1,0,... with g.f. 1 + x + x^5 + x^10 + x^25, where b(.) is the Inverse Euler Transform of A001299.

Examples

			T(30,6)=2 counts the partitions 5+5+5+5+5+5 = 1+1+1+1+1+25.
The triangle starts at row n=0 and has columns k=0..n:
1
0 1
0 0 1
0 0 0 1
0 0 0 0 1
0 1 0 0 0 1
0 0 1 0 0 0 1
0 0 0 1 0 0 0 1
0 0 0 0 1 0 0 0 1
0 0 0 0 0 1 0 0 0 1
0 1 1 0 0 0 1 0 0 0 1
0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 0 1 1 0 0 0 1 0 0 0 1
0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 1 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
0 0 1 1 1 1 2 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1
		

Links

• Alois P. Heinz, Rows n = 0..360, flattened
• Index entries for sequences related to making change.

Crossrefs

Cf. A001299 (row sums), A351740.

Column k=0 gives A000007.

Main diagonal gives A000012.

T(2n,n) gives A351742.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n, b(n, i-1)+
     (p-> `if`(p>n, 0, expand(x*b(n-p, i))))([1, 5, 10, 25][i]))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 4)):
seq(T(n), n=0..15);  # Alois P. Heinz, Feb 17 2022

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, i - 1] +
     Function[p, If[p > n, 0, Expand[x*b[n-p, i]]]][{1, 5, 10, 25}[[i]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 4]];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

Formula

T(n,0) = 0 if k>0.

T(n,n) = 1.

Sum_{k=0..n} k * T(n,k) = A351740(n). - Alois P. Heinz, Feb 17 2022