OFFSET
0,13
COMMENTS
LINKS
Wolfdieter Lang, First 15 rows
Per Alexandersson, Luis Angel González-Serrano, and Egor A. Maximenko, Mario Alberto Moctezuma-Salazar, Symmetric polynomials in the symplectic alphabet and their expression via Dickson-Zhukovsky variables, arXiv:1912.12725 [math.CO], 2019.
Per Alexandersson, Luis Angel González-Serrano, Egor A. Maximenko, and Mario Alberto Moctezuma-Salazar, Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables z_j = x_j + x_j^(-1), The Elec. J. of Combinatorics (2021) Vol. 28, No. 1, #P1.56.
FORMULA
S(1;n,x) = Sum_{k=0..n} S(k,x) = Sum_{m=0..n} a(n,m)*x^m, n >= 0.
a(n,m) = [x^m](S(n,y)*S(n+1,y)/y) with y:=sqrt(2+x).
G.f. for column m: (x^m)/((1-x)*(1+x^2)^(m+1)), which shows that this is a lower diagonal matrix of the Riordan type, named (1/((1+x^2)*(1-x)), x/(1+x^2)).
From Wolfdieter Lang, Oct 16 2012: (Start)
a(n,m) = [x^m](1- (S(n+1,x) - S(n,x)))/(2-x). From the Binet - de Moivre formula for S and use of the geometric sum.
a(n,m) = [x^m](1- R(2*n+3,sqrt(2+x))/sqrt(2+x))/(2-x) with the monic integer T-polynomials R with coefficient triangle given in A127672. From the odd part of the bisection of the T-polynomials. (End)
EXAMPLE
The triangle a(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 10
0: 1
1: 1 1
2: 0 1 1
3: 0 -1 1 1
4: 1 -1 -2 1 1
5: 1 2 -2 -3 1 1
6: 0 2 4 -3 -4 1 1
7: 0 -2 4 7 -4 -5 1 1
8: 1 -2 -6 7 11 -5 -6 1 1
9: 1 3 -6 -13 11 16 -6 -7 1 1
10: 0 3 9 -13 -24 16 22 -7 -8 1 1
... reformatted by Wolfdieter Lang, Oct 16 2012
Row polynomial S(1;4,x) = 1 - x - 2*x^2 + x^3 + x^4 = Sum_{k=0..4} S(k,x).
S(4,y)*S(5,y)/y = 3 - 13*y^2 + 16*y^4 - 7*y^6 + y^8, with y=sqrt(2+x) this becomes S(1;4,x).
From Wolfdieter Lang, Oct 16 2012: (Start)
S(1;4,x) = (1 - (S(5,x) - S(4,x)))/(2-x) = (1-x)*(2-x)*(1+x)*(1-x-x^2)/(2-x) = (1-x)*(1+x)*(1-x-x^2).
S(5,x) - S(4,x) = R(11,sqrt(2+x))/sqrt(2+x) = -1 + 3*x + 3*x^2 - 4*x^3 - x^4 + x^5. (End)
CROSSREFS
Cf. A128495 for S(2; n, x) coefficient table.
KEYWORD
AUTHOR
Wolfdieter Lang, Apr 04 2007
STATUS
approved