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

0,472

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.

Links

Alois P. Heinz, Rows n = 0..360, flattened

Index entries for sequences related to making change.

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

Example

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

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 *)

Crossrefs

Cf. A001299 (row sums), A351740.

Column k=0 gives A000007.

Main diagonal gives A000012.

T(2n,n) gives A351742.

Sequence in context: A134286 A023531 A320841 * A243148 A089495 A345703

Adjacent sequences: A351722 A351723 A351724 * A351726 A351727 A351728

Keyword

nonn,easy,look,tabl

Author

R. J. Mathar, Feb 17 2022

Status

approved