A292047 -id:A292047 - cp's OEIS frontend

A292049 Triangle read by rows: T(n,k) = (-1)^(k-1) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

1, 0, 1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, -2, 0, 1, -2, -1, 0, 1, -3, -1, 0, 1, -3, -2, 0, 1, -4, -3, 0, 1, -4, -4, 1, 0, 1, -5, -5, 1, 0, 1, -5, -7, 2, 0, 1, -6, -8, 3, 0, 1, -6, -10, 5, 0, 1, -7, -12, 6, 1, 0, 1, -7, -14, 9, 1, 0, 1, -8, -16, 11, 2, 0, 1, -8, -19

Offset: 0

Seiichi Manyama, Sep 08 2017

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1, -1;
  0, 1, -1;
  0, 1, -2;
  0, 1, -2, -1;
  0, 1, -3, -1;
  0, 1, -3, -2;
  0, 1, -4, -3;
  0, 1, -4, -4, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give (-1)*A278400.
Columns 0-1 give A000007, A000012.
Cf. A292047.

A292142 Triangle read by rows: T(n,k) = (-4)^(k mod 2) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, -4, 0, -4, 0, -4, -4, 0, -4, -4, 0, -4, -8, 0, -4, -8, 16, 0, -4, -12, 16, 0, -4, -12, 32, 0, -4, -16, 48, 0, -4, -16, 64, 16, 0, -4, -20, 80, 16, 0, -4, -20, 112, 32, 0, -4, -24, 128, 48, 0, -4, -24, 160, 80, 0, -4, -28, 192, 96, -64, 0, -4, -28, 224, 144

Offset: 0

Seiichi Manyama, Sep 09 2017

			First few rows are:
  1;
  0, -4;
  0, -4;
  0, -4,  -4;
  0, -4,  -4;
  0, -4,  -8;
  0, -4,  -8, 16;
  0, -4, -12, 16;
  0, -4, -12, 32;
  0, -4, -16, 48;
  0, -4, -16, 64, 16.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A292135.
Columns 0-1 give A000007, (-4)*A000012.
Cf. A292047.

A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 4, 3, 0, 1, 4, 4, 1, 0, 1, 5, 5, 1, 0, 1, 5, 7, 2, 0, 1, 6, 8, 3, 0, 1, 6, 10, 5, 0, 1, 7, 12, 6, 1, 0, 1, 7, 14, 9, 1, 0, 1, 8, 16, 11, 2, 0, 1, 8, 19, 15, 3, 0, 1, 9, 21, 18, 5, 0, 1, 9, 24, 23, 7

Offset: 0

Seiichi Manyama, Apr 07 2025

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2,  1;
  0, 1, 3,  1;
  0, 1, 3,  2;
  0, 1, 4,  3;
  0, 1, 4,  4, 1;
  0, 1, 5,  5, 1;
  0, 1, 5,  7, 2;
  0, 1, 6,  8, 3;
  0, 1, 6, 10, 5;
  0, 1, 7, 12, 6, 1;
  ...

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A000009.
Columns 0..10 give A000007, A000012, A004526(n-1), A069905(n-3), A026810(n-6), A026811(n-10), A026812(n-15), A026813(n-21), A026814(n-28), A026815(n-36), A026816(n-45).
Cf. A003056, A008284, A291954, A291960, A291968, A292047, A292049.

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1-x^j).

T(n,k) = |A292047(n,k)| = |A292049(n,k)|.

cp's OEIS Frontend

A292049 Triangle read by rows: T(n,k) = (-1)^(k-1) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A292142 Triangle read by rows: T(n,k) = (-4)^(k mod 2) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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