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

Author

Thomas Scheuerle, Jul 07 2022

Keywords

• sign
• tabl

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?

Examples

			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).
		

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).

Programs

PARI

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

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).