A351740 -id:A351740 - cp's OEIS frontend

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

R. J. Mathar, Feb 17 2022

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.

			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

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

Cf. A001299 (row sums), A351740.
Column k=0 gives A000007.
Main diagonal gives A000012.
T(2n,n) gives A351742.

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

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 *)

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

cp's OEIS Frontend

A351725 Table T(n,k) read by rows: number of partitions of n into k parts of size 1, 5, 10 or 25.

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