%I #22 Oct 02 2020 09:45:47

%S 1,1,1,1,1,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,0,0,1,1,

%T 1,1,0,0,1,1,0,1,0,1,1,1,0,1,0,0,1,1,1,1,0,0,1,1,0,1,0,1,1,1,0,1,1,0,

%U 0,0,1,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0,1,1,1,1,1,0,0,0,1,1,0,0

%N Irregular triangle read by rows: T(n,k) is the number of partitions of n into k consecutive parts that differ by 4, and the first element of column k is in the row that is the k-th hexagonal number (A000384).

%C T(n,k) is 0 or 1, so T(n,k) represents the "existence" of the mentioned partition: 1 = exists, 0 = does not exist.

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

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

%C This triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve.

%C For a general theorem about the triangles of this family see A303300.

%e Triangle begins (rows 1..28):

%e 1;

%e 1;

%e 1;

%e 1;

%e 1;

%e 1, 1;

%e 1, 0;

%e 1, 1;

%e 1, 0;

%e 1, 1;

%e 1, 0;

%e 1, 1;

%e 1, 0;

%e 1, 1;

%e 1, 0, 1;

%e 1, 1, 0;

%e 1, 0, 0;

%e 1, 1, 1;

%e 1, 0, 0;

%e 1, 1, 0;

%e 1, 0, 1;

%e 1, 1, 0;

%e 1, 0, 0;

%e 1, 1, 1;

%e 1, 0, 0;

%e 1, 1, 0;

%e 1, 0, 1;

%e 1, 1, 0, 1;

%e ...

%e For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a partition. They are [28], [16, 12] and [13, 9, 5, 1]. The number of parts of these partitions are 1, 2, 4 respectively. There are no partitions of this kind with three parts, so the 28th row of the triangle is [1, 1, 0, 1].

%p A334460 := proc(n, k)

%p local first1 ;

%p first1 := A000384(k) ;

%p if n < first1 then

%p 0 ;

%p elif modp(n-first1, k) = 0 then

%p 1;

%p else

%p 0;

%p end if;

%p end proc:

%p for n from 1 to 40 do

%p for k from 1 do

%p if n>= A000384(k) then

%p printf("%d, ", A334460(n, k)) ;

%p else

%p break;

%p end if;

%p end do:

%p printf("\n") ;

%p end do: # _R. J. Mathar_, Oct 02 2020

%Y Row sums give A334461.

%Y Triangles of the same family where the parts differ by d are A051731 (d=0), A237048 (d=1), A303300 (d=2), A330887 (d=3), this sequence (d=4).

%Y Cf. A000384, A327262, A334462.

%K nonn,tabf

%O 1

%A _Omar E. Pol_, May 01 2020