A263848 Irregular triangle read by rows: row n gives coefficients of basis polynomial {n,k} expressed in terms of binomial coefficients, high order terms first.

1, 1, -1, 1, 0, -1, 1, -1, 1, 1, 0, 0, -1, 2, 0, -1, 1, 2, -1, 0, 1, 1, -1, 1, -1, 1, 0, 0, 0, -1, 3, 0, 0, -1, 1, 5, 0, -1, 0, 1, 3, 0, -1, 1, -1, 3, -1, 0, 0, 1, 5, -2, 0, 1, -1, 3, -2, 1, 0, -1, 1, -1, 1, -1, 1, 1, 0, 0, 0, 0, -1, 4, 0, 0, 0, -1, 1, 9, 0, 0

Offset: 0

Author

N. J. A. Sloane, Nov 15 2015

Keywords

• sign
• tabf
• more

Examples

			Triangle begins:
  1,
  1, -1,
  1,  0, -1,
  1, -1,  1,
  1,  0,  0, -1,
  2,  0, -1,  1,
  2, -1,  0,  1,
  1, -1,  1, -1,
  1,  0,  0,  0, -1,
  3,  0,  0, -1,  1,
  ...
		

Links

• Peter J. C. Moses, First 300 rows.
• Vladimir Shevelev, On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure, arXiv|math.CO/0801.0072, 2007-2010. See Appendix.
• Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1.
• V. Shevelev and J. Spilker, Up-down coefficients for permutations, Elemente der Mathematik, Vol. 68 (2013), no. 3, 115-127.

Crossrefs

Cf. A133457, A264613.

Extensions

More terms from Peter J. C. Moses, Dec 12 2015