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A057094 Coefficient triangle for certain polynomials (rising powers).

%I A057094 #27 Jan 05 2025 19:51:36
%S A057094 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,
%T A057094 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,
%U A057094 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
%N A057094 Coefficient triangle for certain polynomials (rising powers).
%C A057094 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).
%C A057094 Signed version of A284938. - _Eric W. Weisstein_, Apr 06 2017
%H A057094 T. Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>
%H A057094 W. Lang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Note 1 and Table.
%H A057094 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A057094 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.
%e A057094 Triangle begins:
%e A057094 0;
%e A057094 0, -1;
%e A057094 0, 0, -1;
%e A057094 0, 0, 1, -1;
%e A057094 0, 0, 0, 2, -1;
%e A057094 0, 0, 0, -1, 3, -1;
%e A057094 ...
%t A057094 Prepend[CoefficientList[Table[I^n x^(n/2) Fibonacci[n - 1, -I Sqrt[x]], {n, 2, 14}], x], {0}] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%t A057094 Prepend[CoefficientList[Table[-x^(n/2) ChebyshevU[n - 2, Sqrt[x]/2], {n, 2, 14}], x], {0}] // Flatten (* _Eric W. Weisstein_, Apr 06 2017 *)
%o A057094 (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
%Y A057094 Cf. A284938 (unsigned version).
%K A057094 easy,sign,tabl
%O A057094 0,14
%A A057094 _Wolfdieter Lang_, Aug 11 2000