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A057094 Coefficient triangle for certain polynomials (rising powers). 0
0, 0, -1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 2, -1, 0, 0, 0, -1, 3, -1, 0, 0, 0, 0, -3, 4, -1, 0, 0, 0, 0, 1, -6, 5, -1, 0, 0, 0, 0, 0, 4, -10, 6, -1, 0, 0, 0, 0, 0, -1, 10, -15, 7, -1, 0, 0, 0, 0, 0, 0, -5, 20, -21, 8, -1, 0, 0, 0, 0, 0, 0, 1, -15, 35, -28, 9, -1, 0, 0, 0, 0, 0, 0, 0, 6, -35, 56, -36, 10, -1, 0, 0, 0, 0, 0, 0, 0, -1, 21, -70, 84 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are negative scaled Chebyshev U-polynomials: p(n,x)= -U(n-1,sqrt(x)/2)*(sqrt(x))^(n+1), n >= 1. p(0,x)=0. p(n-1,1/x) appears in the n-th power of the g.f. of Catalan's numbers A000108, c(x): (c(x))^n = p(n-1,1/x)*1 -p(n,1/x)*x*c(x). Cf. Lang reference eqs.(1) and (2).

LINKS

Table of n, a(n) for n=0..101.

T. Copeland, Addendum to Elliptic Lie Triad

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Note 1 and Table.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n, m)=0 if n<m; a(0, 0)=0; a(n, m)= ((-1)^(n-m+1))*binomial(m-1, n-m) if n >= 1 and n >= m >=floor(n/2)+1; else 0.

EXAMPLE

Triangle begins:

0;

0, -1;

0, 0, -1;

0, 0, 1, -1;

0, 0, 0, 2, -1;

0, 0, 0, -1, 3, -1;

...

PROG

(PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, if ((n==0) || (k < n\2+1), v = 0, v = (-1)^(n-k+1)*binomial(k-1, n-k)); print1(v, ", "); ); print(); ); } \\ Michel Marcus, Jan 14 2016

CROSSREFS

Sequence in context: A110174 A022909 A032239 * A186084 A047998 A017847

Adjacent sequences:  A057091 A057092 A057093 * A057095 A057096 A057097

KEYWORD

easy,sign,tabl

AUTHOR

Wolfdieter Lang, Aug 11 2000

STATUS

approved

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Last modified December 7 14:52 EST 2016. Contains 278877 sequences.