A027478 Triangle of the cube of the normalized, unsigned Stirling matrix of the first kind.

1, 7, 1, 176, 39, 1, 10746, 2951, 126, 1, 1297704, 407450, 22535, 310, 1, 272866980, 94128364, 6139575, 112435, 645, 1, 91570835040, 33910601508, 2487385684, 54814095, 426475, 1197, 1, 46034917019280, 18030534782364, 1446119232796, 36402686929, 345710680, 1333906, 2044, 1

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• tabl
• easy

Comments

The absolute values are unchanged if one uses the signed Stirling numbers of the first kind.

Examples

			The first rows of the triangle are :
  1,
  7, 1,
  176, 39, 1,
  10746, 2951, 126, 1,
  1297704, 407450, 22535, 310, 1,
  272866980, 94128364, 6139575, 112435, 645, 1,
  ...
		

Crossrefs

Cf. A027477 for the quadratic version.

Cf. A027479 for the quartic version.

Cf. A027482 is the first subdiagonal of this triangle.

Programs

Mathematica

Module[{nmax=8,m},m=(Table[Table[(-1)^(i+j) StirlingS1[i,j]/i!,{j,1,nmax}],{i,1,nmax}]);m=m.m.m*Table[i!^3,{i,1,nmax}]; Flatten[Table[Table[m[[i,j]],{j,1,i}],{i,1,nmax}],1]]

Formula

Let A be the lower triangular matrix with entries a[ i, j ] = (-1)^(i+j)*s(i, j)/i! if j<=i, 0 if j>i, where s(i,j) is the Stirling number of the first kind. Let N be the column vector ((i!^3)).

T is the lower triangular matrix A.A.A.N.

Extensions

Definition, formula and program edited for clarity by Olivier Gérard, Jan 20 2019