A330888 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 3, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th pentagonal number (A000326).

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 4, 1, 2, 0, 0, 1, 0, 3, 0, 1, 2, 0, 0, 1, 0, 0, 4, 1, 2, 3, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 3, 4, 1, 2, 0, 0, 1, 0, 0, 0, 1, 2, 3, 0, 1, 0, 0, 4, 1, 2, 0, 0, 5

Offset

1,6

Comments

Since the trivial partition n is counted, so T(n,1) = 1.

This is an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in the row that is the k-th pentagonal number.

Links

Table of n, a(n) for n=1..105.

Formula

T(n,k) = k*A330887(n,k).

Example

Triangle begins (rows 1..26):
1;
1;
1;
1;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
1, 0, 0;
1, 2, 0;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
1, 0, 0, 4;
1, 2, 0, 0;
1, 0, 3, 0;
1, 2, 0, 0;
1, 0, 0, 4;
...
For n = 21 there are three partitions of 21 into consecutive parts that differ by 3, including 21 as a partition. They are [21], [12, 9] and [10, 7, 4]. The number of parts of these partitions are 1, 2 and 3 respectively, so the 21st row of the triangle is [1, 2, 3].

Maple

A330888 := proc(n, k)
    k*A330887(n, k) ;
end proc:
for n from 1 to 40 do
    for k from 1 do
        if n>= A000325(k) then
            printf("%d, ", A330888(n, k)) ;
        else
            break;
        end if;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Oct 02 2020

Crossrefs

Cf. A000326, A330887.

Other triangles of the same family are A127093, A285914, A330466.

Sequence in context: A054848 A284171 A286320 * A194525 A330466 A282938

Adjacent sequences: A330885 A330886 A330887 * A330889 A330890 A330891

Keyword

nonn,tabf

Author

Omar E. Pol, Apr 30 2020

Status

approved