A395066 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = [x^k] Product_{j=0..n} (1 + j^4*x).

1, 1, 1, 1, 17, 16, 1, 98, 1393, 1296, 1, 354, 26481, 357904, 331776, 1, 979, 247731, 16908529, 224021776, 207360000, 1, 2275, 1516515, 337967905, 22137475360, 290539581696, 268738560000, 1, 4676, 6978790, 3979120420, 833598415265, 53442617921056, 697854274212096, 645241282560000

Offset

0,5

Links

Seiichi Manyama, Rows n = 0..139, flattened

Formula

T(n,k) = T(n-1,k) + n^4 * T(n-1,k-1).

T(n,k) = Sum_{1 <= x_1 < x_2 < ... < x_{n-k} <= n} ( n!/Product_{j=1..n-k} x_j )^4.

T(n,k) = Sum_{1 <= x_1 < x_2 < ... < x_k <= n} ( Product_{j=1..k} x_j )^4.

Example

Triangle begins:
  1;
  1,   1;
  1,  17,     16;
  1,  98,   1393,     1296;
  1, 354,  26481,   357904,    331776;
  1, 979, 247731, 16908529, 224021776, 207360000;
  ...

Prog

(PARI) T(n, k) = if(n*k==0, n^k, T(n-1, k)+n^4*T(n-1, k-1));

Crossrefs

Columns k=0..2 give A000012, A000538, A351760.

Diagonals give A134375, A203229, A395068, A395897.

Row sums give A255434.

Cf. A008955, A094638, A249677.

Sequence in context: A085095 A004506 A096181 * A248941 A261306 A022973

Adjacent sequences: A395063 A395064 A395065 * A395067 A395068 A395069

Keyword

nonn,tabl

Author

Seiichi Manyama, May 09 2026

Status

approved