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

1,2

Comments

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

Links

Table of n, a(n) for n=1..36.

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.

Example

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,
...

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]]

Crossrefs

Cf. A027477 for the quadratic version.

Cf. A027479 for the quartic version.

Cf. A027482 is the first subdiagonal of this triangle.

Sequence in context: A052122 A027538 A334394 * A009792 A103243 A027496

Adjacent sequences: A027475 A027476 A027477 * A027479 A027480 A027481

Keyword

nonn,tabl,easy

Author

Olivier Gérard

Extensions

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

Status

approved