A331900 Number of compositions (ordered partitions) of the n-th triangular number into distinct triangular numbers.

1, 1, 1, 1, 7, 1, 3, 13, 3, 55, 201, 159, 865, 1803, 7093, 43431, 14253, 22903, 130851, 120763, 1099693, 4527293, 4976767, 7516897, 14349685, 72866239, 81946383, 167841291, 897853735, 455799253, 946267825, 5054280915, 3941268001, 17066300985, 49111862599

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Index entries for sequences related to compositions

Formula

a(n) = A331843(A000217(n)).

Example

a(6) = 3 because we have [21], [15, 6] and [6, 15].

Maple

b:= proc(n, i, p) option remember; (t->
      `if`(t*(i+2)/3<n, 0, `if`(n=0, p!, b(n, i-1, p)+
      `if`(t>n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
    end:
a:= n-> b(n*(n+1)/2, n, 0):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 31 2020

Mathematica

b[n_, i_, p_] := b[n, i, p] = With[{t = i(i+1)/2}, If[t(i+2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t > n, 0, b[n-t, i-1, p+1]]]]];
a[n_] := b[n(n+1)/2, n, 0];
a /@ Range[0, 37] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000217, A072964, A126683, A196010, A224677, A288126, A298858, A307614, A331843, A331884.

Sequence in context: A201585 A089562 A104621 * A176439 A048834 A010504

Adjacent sequences: A331897 A331898 A331899 * A331901 A331902 A331903

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 31 2020

Extensions

More terms from Alois P. Heinz, Jan 31 2020

Status

approved