A133728 -id:A133728 - cp's OEIS frontend

A133729 A007318 * A133728.

1, 1, 3, 2, 6, 5, 4, 12, 15, 7, 8, 24, 35, 28, 9, 16, 48, 75, 77, 45, 11, 32, 96, 155, 182, 144, 66, 13, 64, 192, 315, 399, 378, 242, 91, 15, 128, 384, 635, 840, 891, 704, 377, 120, 17

Offset: 1

Gary W. Adamson, Sep 21 2007

Row sums = A049611: (1, 4, 13, 38, 104, 272, ...).

			First few rows of the triangle:
   1;
   1,  3;
   2,  6,  5;
   4, 12, 15,  7;
   8, 24, 35, 28,  9;
  16, 48, 75, 77, 45, 11;
  ...

Cf. A133728, A049611.

Binomial transform of triangle A133728.

A354596 Array T(n,k) = k^2 + (2n-4)*floor(k/2)^2, n >= 0, k >= 0, read by descending antidiagonals.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 5, 2, 1, 0, 0, 7, 4, 1, 0, 9, 8, 9, 6, 1, 0, 0, 17, 16, 11, 8, 1, 0, 13, 18, 25, 24, 13, 10, 1, 0, 0, 31, 36, 33, 32, 15, 12, 1, 0, 17, 32, 49, 54, 41, 40, 17, 14, 1, 0, 0, 49, 64, 67, 72, 49, 48, 19, 16, 1, 0, 21, 50, 81, 96, 85, 90, 57, 56, 21, 18, 1, 0

Offset: 0

David Lovler, Jun 01 2022

Column k is an arithmetic progression with difference 2*A008794(k).
Odd rows of A133728 triangle are contained in row 0.
For i = 0 through 4, row i is 0 and the diagonal of A319929, A322630 = A213037, A003991, A322744, and A327259, respectively. In general, row i is 0 and the diagonal of array U(i;n,k) described in A327263.

			T(n,k) begins:
  0,   1,   0,   5,   0,   9,   0,  13, ...
  0,   1,   2,   7,   8,  17,  18,  31, ...
  0,   1,   4,   9,  16,  25,  36,  49, ...
  0,   1,   6,  11,  24,  33,  54,  67, ...
  0,   1,   8,  13,  32,  41,  72,  85, ...
  0,   1,  10,  15,  40,  49,  90, 103, ...
  0,   1,  12,  17,  48,  57, 108, 121, ...
  ...

David Lovler, Table of n, a(n) for n = 0..5150

Cf. A000290, A008794, A133728, A213037, A247375.
Cf. A266222, A266439.
Cf. A319929, A322630, A322744, A327259, A327263, A354594, A354595.

T[n_, k_] := k^2 + (2*n - 4)*Floor[k/2]^2; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jun 20 2022 *)

PARI
```
T(n,k) = k^2 + (2*n-4)*(k\2)^2;
```

T(n,k) = U(n;k,k) (see A327263).
For each row, T(n,k) = T(n,k-1) + 2*T(n,k-2) - 2*T(n,k-3) - T(n,k-4) + T(n,k-5), k >= 5.
G.f. for row n: x*(1 + (2*n-1)*x + 3*x^2 + (2*n-3)*x^3)/((1 - x)^3*(1 + x)^2). When n = 2, this reduces to x*(1 + x)/(1 - x)^3.
E.g.f. for row n: (((4-n)*x + n*x^2)*cosh(x) + (n-2 + n*x + n*x^2)*sinh(x))/2. When n = 2, this reduces to (x + x^2)*cosh(x) + (x + x^2)*sinh(x) = (x + x^2)*exp(x).

cp's OEIS Frontend

A133729 A007318 * A133728.

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A354596 Array T(n,k) = k^2 + (2n-4)*floor(k/2)^2, n >= 0, k >= 0, read by descending antidiagonals.

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