A291929 -id:A291929 - cp's OEIS frontend

A291930 Row sums of A291929.

1, 1, 2, 5, 11, 26, 60, 139, 321, 743, 1716, 3965, 9158, 21152, 48848, 112808, 260500, 601553, 1389096, 3207660, 7406989, 17103860, 39495306, 91200333, 210594475, 486292240, 1122916743, 2592971247, 5987531168, 13826041086, 31926247578, 73722134145, 170234630412

Offset: 0

Seiichi Manyama, Sep 06 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2752

Cf. A291896, A291905.

A291904 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 3, 2, 0, 0, 3, 2, 1, 1, 1, 0, 2, 3, 2, 1, 0, 0, 3, 4, 3, 1, 0, 0, 4, 4, 3, 2, 1, 0, 4, 6, 4, 2, 0, 0, 6, 7

Offset: 0

Seiichi Manyama, Sep 05 2017

T(n,k) is the number of integer compositions of n with first part 1, last part k, and all adjacent differences in {-1,1}. - John Tyler Rascoe, Aug 14 2023

			First few rows are:
  1;
  0, 1;
  0, 0;
  0, 0, 1;
  0, 1, 0;
  0, 0, 0;
  0, 0, 1, 1;
  0, 1, 0, 0;
  0, 0, 1, 0;
  0, 1, 1, 1;
  0, 1, 0, 0, 1;
  0, 0, 2, 1, 0;
  0, 2, 1, 1, 0.

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

Row sums give A291905.
Columns 0-1 give A000007, A227310 (for n>0).
Cf. A008289, A173258, A291895, A291929.

T[0, 0] = 1; T[, 0] = 0; T[n?Positive, k_] /; 0 < k <= Floor[(Sqrt[8n+1] - 1)/2] := T[n, k] = T[n-k, k-1] + T[n-k, k+1]; T[, ] = 0;
Table[T[n, k], {n, 0, 20}, {k, 0, Floor[(Sqrt[8n+1] - 1)/2]}] // Flatten (* Jean-François Alcover, May 29 2019 *)

From John Tyler Rascoe, Aug 14 2023: (Start)
This triangle is T_1(n,k) of the general triangle T_m(n,k) for compositions of this kind with first part m.
T_m(n,k) for 0 < m, 0 <= n, and 0 <= k <= A003056(n+A000217(m-1)).
T_m(0,0) = T_m(m,m) = 1.
T_m(n,k) = T_m(n-k,k-1) + T_m(n-k,k+1) for m < n and 0 < k <= A003056(n+A000217(m-1)).
T_m(n,k) = 0 for 0 < n < m or n < k.
T_m(n,0) = 0 for 0 < n. (End)

A291954 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, 0, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, -1, 0, 0, 1, 0, -1, 0, -1, 0, 2, 1, 0, 1, 1, -1, -1, 0, -1, -1, 1, 0, 0, 1, 0, -2, -1, 0, -1, 0, 2, 1, 0, 1, 1, -2, 0, 1, 0, -1, -1, 2, 1, -1, 0, 1, 0, -2, -1, 0, 0, -1, 0, 3, 1, -1, 0, 1

Offset: 0

Seiichi Manyama, Sep 06 2017

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

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

Row sums give A003406.
Columns 0-1 give A000007, A062157.
Cf. A008289, A291895, A291929.

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

A291940 Triangle read by rows: T(n,k) = T(n-k,k-1) - 2*T(n-k,k) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, -2, 0, 4, 1, 0, -7, -2, 0, 12, 2, 0, -22, -3, 1, 0, 41, 8, -2, 0, -74, -15, 2, 0, 133, 23, -5, 0, -243, -42, 12, 1, 0, 444, 82, -19, -2, 0, -806, -147, 33, 2, 0, 1465, 261, -65, -5, 0, -2669, -479, 118, 10, 0, 4859, 878, -211, -15, 1, 0, -8840, -1593, 386

Offset: 0

Seiichi Manyama, Sep 06 2017

			First few rows are:
  1;
  0,    1;
  0,   -2;
  0,    4,   1;
  0,   -7,  -2;
  0,   12,   2;
  0,  -22,  -3,  1;
  0,   41,   8, -2;
  0,  -74, -15,  2;
  0,  133,  23, -5;
  0, -243, -42, 12, 1.

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

Row sums give A291942.
Columns 0-1 give A000007, (-1)*A275762 (for n>0).
Cf. A008289, A291895, A291929.

cp's OEIS Frontend

A291930 Row sums of A291929.

Views

Author

Keywords

Links

Crossrefs

A291904 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A291954 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).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A291940 Triangle read by rows: T(n,k) = T(n-k,k-1) - 2*T(n-k,k) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

Views

Author

Keywords

Examples

Links

Crossrefs