A349226 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n} (x - k^k) expanded in decreasing powers of x, with row 0 = {1}.

1, 1, -1, 1, -2, 1, 1, -6, 9, -4, 1, -33, 171, -247, 108, 1, -289, 8619, -44023, 63340, -27648, 1, -3413, 911744, -26978398, 137635215, -197965148, 86400000, 1, -50070, 160195328, -42565306462, 1258841772303, -6421706556188, 9236348345088, -4031078400000

Offset

0,5

Comments

Let M be an n X n matrix filled by binomial(i*j, i) with rows and columns j = 1..n, k = 1..n; then its determinant equals unsigned T(n, n). Can we find a general formula for T(n+m, n) based on determinants of matrices and binomials?

Links

Table of n, a(n) for n=0..35.

Formula

T(n, 0) = 1.

T(n, 1) = -A062970(n).

T(n, 2) = Sum_{m=0..n-1} A062970(m)*m^m.

T(n, k) = Sum_{m=0..n-1} -T(m, k-1)*m^m.

T(n, n) = (-1)^n*A002109(n).

Example

The triangle begins:
  1;
  1,    -1;
  1,    -2,      1;
  1,    -6,      9,        -4;
  1,   -33,    171,      -247,       108;
  1,  -289,   8619,    -44023,     63340,     -27648;
  1, -3413, 911744, -26978398, 137635215, -197965148, 86400000;
  ...
Row 4: x^4-33*x^3+171*x^2-247*x+108 = (x-1)*(x-1^1)*(x-2^2)*(x-3^3).

Prog

(PARI) T(n, k) = polcoeff(prod(m=0, n-1, (x-m^m)), n-k);

Crossrefs

Cf. A002109, A062970.

Cf. A008276 (The Stirling numbers of the first kind in reverse order).

Cf. A039758 (Coefficients for polynomials with roots in odd numbers).

Cf. A355540 (Coefficients for polynomials with roots in factorials).

Sequence in context: A196073 A144089 A172107 * A165891 A039763 A094262

Adjacent sequences: A349223 A349224 A349225 * A349227 A349228 A349229

Keyword

sign,tabl

Author

Thomas Scheuerle, Jul 07 2022

Status

approved