A389524 Triangle read by rows. T(1, 1) = 1, T(n, k) = [n >= k](1 + Sum_{i=1..k-1} T(n - i, k - 1) - Sum_{i=1..n-1} T(n - i, k)).

1, 0, 2, 0, -1, 3, 0, 0, -1, 4, 0, 0, -2, -1, 5, 0, 0, 1, -2, -1, 6, 0, 0, 0, -2, -2, -1, 7, 0, 0, 0, 1, -2, -2, -1, 8, 0, 0, 0, 2, -3, -2, -2, -1, 9, 0, 0, 0, -1, 3, -3, -2, -2, -1, 10, 0, 0, 0, 0, 1, -2, -3, -2, -2, -1, 11, 0, 0, 0, 0, 2, 2, -2, -3, -2, -2, -1, 12

Offset

1,3

Comments

Conjecture 1: A000005(n) = Sum_{k >= 1} T(n, k), verified up to n = 1000.

Conjecture 2: A274628(n) = Sum_{k >= 1} T(n, n - k + 1)*(n - k + 1), verified up to n = 500.

Conjecture 3: A049820(n) = T(2*n, n), verified up to n = 250.

A389526(n) = Sum_{k >= 1} T(n, 2*k - 1).

A387793(n) = Sum_{k >= 1} T(n, n - 2*k + 1).

A389573(n) = Sum_{k >= 1} T(n, n - 2*k + 2).

Links

Table of n, a(n) for n=1..78.

Example

Triangle begins:
  {1},
  {0,  2},
  {0, -1,  3},
  {0,  0, -1,  4},
  {0,  0, -2, -1,  5},
  {0,  0,  1, -2, -1,  6},
  {0,  0,  0, -2, -2, -1,  7},
  {0,  0,  0,  1, -2, -2, -1,  8},
  {0,  0,  0,  2, -3, -2, -2, -1,  9},
  {0,  0,  0, -1,  3, -3, -2, -2, -1, 10},
  {0,  0,  0,  0,  1, -2, -3, -2, -2, -1, 11},
  {0,  0,  0,  0,  2,  2, -2, -3, -2, -2, -1, 12}

Mathematica

Clear[t]; nn = 12; t[n_, 1] = If[n == 1, 1, 0]; t[n_, k_] := t[n, k] = If[n >= k, (1 + Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, n - 1}]), 0]; Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]]

Prog

(PARI) tabl(nn) = my(m = matrix(nn, nn, i, j, oo)); for (n=1, nn, for (k=1, nn, m[n, k] = if (n==1, if (k==1, 1), if (n>=k, 1 + sum(i=1, k-1, m[n-i, k-1]) - sum(i=1, n-1, m[n-i, k]), 0)); ); ); m; \\ Michel Marcus, Oct 16 2025

Crossrefs

Cf. A380385 (recurrence), A000005, A025052, A274628, A389526, A387793, A389573.

Sequence in context: A127701 A158821 A004199 * A062283 A136493 A338849

Adjacent sequences: A389521 A389522 A389523 * A389525 A389526 A389527

Keyword

sign,tabl

Author

Mats Granvik, Oct 07 2025

Status

approved