A309559 -id:A309559 - cp's OEIS frontend

A309557 Number triangle with T(n,k) = 2 + 3n - 2k + 2k(n-k) for n >= 0, 0 <= k <= n.

2, 5, 3, 8, 8, 4, 11, 13, 11, 5, 14, 18, 18, 14, 6, 17, 23, 25, 23, 17, 7, 20, 28, 32, 32, 28, 20, 8, 23, 33, 39, 41, 39, 33, 23, 9, 26, 38, 46, 50, 50, 46, 38, 26, 10, 29, 43, 53, 59, 61, 59, 53, 43, 29, 11, 32, 48, 60, 68, 72, 72, 68, 60, 48, 32, 12, 35, 53, 67, 77, 83, 85, 83, 77, 67, 53, 35, 13

Offset: 0

Philip K Hotchkiss, Aug 07 2019

The rascal triangle (A077028) can be generated by South = (East*West+1)/North or South = East+West+1-North; this triangle can be generated by South = (East*West+1)/North, South = East+West+2-North.

			For row n=3: a(3,0)=11, a(3,1)=13, a(3,2)=11, a(3,3)=5, ...
For antidiagonal r=2: T(2,0)=4, T(2,1)=11, T(2,2)=18, ...
Triangle T begins:
              2
            5   3
          8   8   4
        11  13  11  5
      14  18  18  14  6
    17  23  25  23  17  7
  20  28  32  32  28  20  8
23  33  39  41  39  33  23  9
             ...

Philip K Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019.

Cf. A077028, A309555, A309559.

:=proc(n,k)
   if n<0 or k<0 or k>n then
       0;
   else
       2+3*n -2*k +2*k*(n-k);
   end if;

T[n_,k_]:=2+3*n-2*k+2*k*(n-k); Table[T[n,k], {n,0,11}, {k,0,n}] // Flatten
f[n_] := Table[SeriesCoefficient[(x*(-1-4*y+y^2)-2*(1-4*y+y^2))/((-1+x)^2*(-1+y)^3), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Flatten[Array[f, 12,0]] (* Stefano Spezia, Sep 08 2019 *)

By rows: a(n,k) = 2 + 3*n - 2*k + 2*k*(n-k) n >= 0, 0 <= k <= n.
By antidiagonals: T(r,k) = 2 + 3*k + r + 2*r*k, r,k >= 0.
G.f.: (x*(-1-4*y+y^2)-2*(1-4*y+y^2))/((-1+x)^2*(-1+y)^3). - Stefano Spezia, Sep 08 2019

A332790 Triangle read by rows: T(n,k) = 1 + 2n + k + 5k(n-k) for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 3, 4, 5, 11, 7, 7, 18, 19, 10, 9, 25, 31, 27, 13, 11, 32, 43, 44, 35, 16, 13, 39, 55, 61, 57, 43, 19, 15, 46, 67, 78, 79, 70, 51, 22, 17, 53, 79, 95, 101, 97, 83, 59, 25, 19, 60, 91, 112, 123, 124, 115, 96, 67, 28, 21, 67, 103, 129, 145, 151, 147, 133, 109, 75, 31

Offset: 0

Philip K Hotchkiss, Mar 04 2020

			From _Jon E. Schoenfield_, Mar 14 2020: (Start)
.
  n\k|  0    1    2    3    4    5    6    7    8    9   10
  ---+-----------------------------------------------------
   0 |  1
   1 |  3    4
   2 |  5   11    7
   3 |  7   18   19   10
   4 |  9   25   31   27   13
   5 | 11   32   43   44   35   16
   6 | 13   39   55   61   57   43   19
   7 | 15   46   67   78   79   70   51   22
   8 | 17   53   79   95  101   97   83   59   25
   9 | 19   60   91  112  123  124  115   96   67   28
  10 | 21   67  103  129  145  151  147  133  109   75   31
  ...
(End)

Philip K. Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019, Figure 8 p. 3.

Cf. A077028, A309555, A309557, A309559, A332963

:=proc(n, k)
   if n<0 or k<0 or k>n then
       0;
   else
       1+2*n+k+5*k*(n-k);
   end if;

T[n_, k_]:=1+2*n+k+5*k*(n-k); Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

T(n,k) = 1 + 2*n + k + 5*k*(n-k), n >= 0, 0 <= k <= n.

A332963 Number triangle where T(2n,0)=T(2n,2n)=1, T(2n+1,0)=T(2n+1,2n+1)=2 for all n >= 0, and the interior numbers are defined recursively by T(n,k) = (T(n-1,k-1)*T(n-1,k)+1)/T(n-2,k-1) for n > 2, 0 < k <= n.

Original entry on oeis.org

1, 2, 2, 1, 5, 1, 2, 3, 3, 2, 1, 7, 2, 7, 1, 2, 4, 5, 5, 4, 2, 1, 9, 3, 13, 3, 9, 1, 2, 5, 7, 8, 8, 7, 5, 2, 1, 11, 4, 19, 5, 19, 4, 11, 1, 2, 6, 9, 11, 12, 12, 11, 9, 6, 2, 1, 13, 5, 25, 7, 29, 7, 25, 5, 13, 1, 2, 7, 11, 14, 16, 17, 17, 16, 14, 11, 7, 2

Offset: 0

Philip K Hotchkiss, Mar 04 2020

			For row 3: a(3,0)=2, a(3,1)= 3, a(3,2)=3, a(3,3)=2.
For antidiagonal 3: T(3,0)=2, T(3,1)=7, T(3,2)=5, T(3,3)=13, ...
Triangle begins:
  1;
  2, 2;
  1, 5, 1;
  2, 3, 3, 2;
  1, 7, 2, 7, 1;
  2, 4, 5, 5, 4, 2;
  ...

Philip K. Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019, Figure 10 p. 4.

Cf. A077028, A309555, A309557, A332790, A309559

T(n, k) = if ((n<0) || (nMichel Marcus, Mar 16 2020

By rows: a(2n,0)=a(2n,2n)=1, a(2n+1,0)=a(2n+1,2n+1)=2 for all n >= 0, while the interior numbers are defined recursively by a(n,k) = (a(n-1,k-1)*a(n-1,k)+1)/a(n-2,k-1) for n >= 2, 0 < k <= n.

By antidiagonals: T(0,2n)=T(2n,0)=1, T(0,2n+1)=T(2n+1,0)=2 for all n >= 0, while the interior numbers are defined recursively by T(r,k) = (T(r-1,k)*(Tr,k-1)+1)/T(r-1,k-1) for r,k > 0.

A329854 Triangle read by rows: T(n,k) = ((n - k)*(n + k - 1) + 2)/2, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 7, 6, 4, 1, 11, 11, 10, 8, 5, 1, 16, 16, 15, 13, 10, 6, 1, 22, 22, 21, 19, 16, 12, 7, 1, 29, 29, 28, 26, 23, 19, 14, 8, 1, 37, 37, 36, 34, 31, 27, 22, 16, 9, 1, 46, 46, 45, 43, 40, 36, 31, 25, 18, 10, 1, 56, 56, 55, 53, 50, 46, 41, 35, 28, 20, 11, 1

Offset: 0

Werner Schulte, Nov 22 2019

This triangle equals A309559 with reversed rows and supplemented main diagonal (all terms are 1).
There are two lower triangular matrices M and N so that the matrix product M * N equals T (seen as a matrix).
           / 1             \                 / 1             \
           | 0 1           |                 | 1 1           |
           | 0 1 1         |                 | 1 1 1         |
  M(n,k) = | 0 1 2 1       |        N(n,k) = | 1 1 1 1       |
           | 0 1 2 3 1     |                 | 1 1 1 1 1     |
           | 0 1 2 3 4 1   |                 | 1 1 1 1 1 1   |
           \ . . . . . . . /                 \ . . . . . . . /
  The matrix product N * M equals the rascal triangle A077028 (seen as a matrix).

			The triangle T(n,k) starts:
n \ k :   0    1    2    3    4    5    6    7    8    9   10   11
==================================================================
   0  :   1
   1  :   1    1
   2  :   2    2    1
   3  :   4    4    3    1
   4  :   7    7    6    4    1
   5  :  11   11   10    8    5    1
   6  :  16   16   15   13   10    6    1
   7  :  22   22   21   19   16   12    7    1
   8  :  29   29   28   26   23   19   14    8    1
   9  :  37   37   36   34   31   27   22   16    9    1
  10  :  46   46   45   43   40   36   31   25   18   10    1
  11  :  56   56   55   53   50   46   41   35   28   20   11    1
etc.

Row sums equal A116731(n+1).
Row sums apart from column 0 equal A081489.
Cf. A077028, A309559.

O.g.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = ((t^2+(1-t)^2) * (1-x*t) + x * t^2 * (1-t)) / ((1-t)^3 * (1-x*t)^2).
G.f. of column k: Sum_{n>=k} T(n,k) * t^n = t^k * (t^2/(1-t)^3 + 1/(1-t) + k*t/(1-t)^2) for k >= 0.
T(n,k) = 1 + T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 0 < k < n with initial values T(n,0) = (n*(n-1)+2)/2 and T(n,n) = 1 for n >= 0.
T(n,k) = (2 + T(n-1,k-1) * T(n-1,k+1)) / T(n-2,k) for 0 < k < n-1 with initial values given above and T(n,n-1) = n for n > 0.
Referring to the triangle M(n,k) (see comments), we get:
  (1) Sum_{k=0..n} (k+1) * M(n,k) = A116731(n+1) for n >= 0;
  (2) Sum_{k=1..n} k * M(n,k) = A081489(n) for n >= 1.
T(n,k) = T(n-1,k-1) + n-k for 0 < k <= n with initial values T(n,0) = (n*(n-1)+2)/2 for n >= 0.
T(n,k) = 2 * T(n-1,k-1) - T(n-2,k-2) for 1 < k <= n with initial values T(0,0) = 1 and T(n,0) = T(n,1) = (n*(n-1)+2)/2 for n > 0.

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A309557 Number triangle with T(n,k) = 2 + 3*n - 2*k + 2*k*(n-k) for n >= 0, 0 <= k <= n.

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A332790 Triangle read by rows: T(n,k) = 1 + 2*n + k + 5*k(n-k) for n >= 0, 0 <= k <= n.

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A332963 Number triangle where T(2n,0)=T(2n,2n)=1, T(2n+1,0)=T(2n+1,2n+1)=2 for all n >= 0, and the interior numbers are defined recursively by T(n,k) = (T(n-1,k-1)*T(n-1,k)+1)/T(n-2,k-1) for n > 2, 0 < k <= n.

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A329854 Triangle read by rows: T(n,k) = ((n - k)*(n + k - 1) + 2)/2, 0 <= k <= n.

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A309557 Number triangle with T(n,k) = 2 + 3n - 2k + 2k(n-k) for n >= 0, 0 <= k <= n.

A332790 Triangle read by rows: T(n,k) = 1 + 2n + k + 5k(n-k) for n >= 0, 0 <= k <= n.