A355570 Regular triangle of certain polynomial expansion coefficients for the n-th power series.

1, 0, 1, 1, -2, 2, 0, 5, -10, 6, 1, -10, 40, -54, 24, 0, 21, -140, 336, -336, 120, 1, -42, 462, -1764, 3024, -2400, 720, 0, 85, -1470, 8442, -22176, 29520, -19440, 5040, 1, -170, 4580, -38178, 144648, -288000, 313200, -176400, 40320, 0, 341, -14080, 166452, -875952, 2451240, -3920400, 3603600, -1774080, 362880

Offset: 2

Author

Michel Marcus, Jul 07 2022

Keywords

• sign
• tabl

Comments

See the paper by Muschielok for precise definition.

Examples

			Triangle begins:
  1;
  0,   1;
  1,  -2,    2;
  0,   5,  -10,   6;
  1, -10,   40, -54,   24;
  0,  21, -140, 336, -336, 120;
  ...
		

Links

• Michel Marcus, Table of n, a(n) for n = 2..4951 (Rows 2..100)
• Christoph Muschielok, Another Approach on Power Sums, arXiv:2207.01935 [math.CO], 2022.
• Christoph Muschielok, Further Identities for cmk- and amk-Weighted Sums and a Remark on a Representation of Pythagoras' Equation, arXiv:2208.11032 [math.NT], 2022.

Crossrefs

Cf. A000142 (right diagonal), A202365 (subdiagonal).

Programs

PARI

mat(n) = my(M = matrix(n, n)); M[1, 1] = 1; for (i=2, n, my(p=x + prod(k=-1, i-2, x+k)/(i-2)!); for (j=1, i, M[i, j] = polcoef(p, j, x));); my(iM = 1/M); matrix(n-1, n-1, i, j, iM[i+1, j+1]);
tabl(nn) = {my(m = mat(nn)); for (n=1, nn-1, row = vector(n, k, m[n, k]); print(row, ", "););}