A249677 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^3*x).

1, 1, 1, 1, 9, 8, 1, 36, 251, 216, 1, 100, 2555, 16280, 13824, 1, 225, 15055, 335655, 2048824, 1728000, 1, 441, 63655, 3587535, 74550304, 444273984, 373248000, 1, 784, 214918, 25421200, 1305074809, 26015028256, 152759224512, 128024064000, 1, 1296, 616326, 135459216, 14320729209, 694213330464, 13472453691584, 78340747014144, 65548320768000

Offset

0,5

Comments

Column 1 forms the squares of the triangular numbers (A000537).

Main diagonal forms the cubes of the factorial numbers (A000442).

Row sums equal Product_{k=1..n} (k^3 + 1) = n!*Product_{k=1..n} (k*(k-1) + 1) = n!*A130032(n).

Links

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

Formula

From Seiichi Manyama, May 09 2026: (Start)

T(n,k) = T(n-1,k) + n^3 * 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 )^3.

T(n,k) = Sum_{1 <= x_1 < x_2 < ... < x_k <= n} ( Product_{j=1..k} x_j )^3. (End)

Example

Triangle begins:
  1;
  1,   1;
  1,   9,      8;
  1,  36,    251,      216;
  1, 100,   2555,    16280,      13824;
  1, 225,  15055,   335655,    2048824,     1728000;
  1, 441,  63655,  3587535,   74550304,   444273984,    373248000;
  1, 784, 214918, 25421200, 1305074809, 26015028256, 152759224512, 128024064000;
  ...

Prog

(PARI) {T(n, k)=polcoeff(prod(m=0, n, 1 + m^3*x +x*O(x^n)), k)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Columns k=0..3 give A000012, A000537, A347107, A352980.

Diagonals give A000442, A066989, A394716, A394861.

Row sums give A255433.

Cf. A008955, A094638, A395066.

Cf. A107415, A130032.

Sequence in context: A185260 A275615 A385962 * A094135 A021897 A378205

Adjacent sequences: A249674 A249675 A249676 * A249678 A249679 A249680

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 03 2014

Status

approved