A182867 Triangle read by rows: row n gives coefficients in expansion of Product_{i=1..n} (x - (2i)^2), highest powers first.

1, 1, -4, 1, -20, 64, 1, -56, 784, -2304, 1, -120, 4368, -52480, 147456, 1, -220, 16368, -489280, 5395456, -14745600, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400, 1, -560, 119392, -12263680, 633721088, -15658639360, 157294854144, -416179814400, 1, -816, 262752, -42828032, 3773223168, -177891237888, 4165906530304, -40683662475264, 106542032486400, 1, -1140, 527136, -127959680, 17649505536, -1400415544320, 61802667606016, -1390437378293760, 13288048674471936, -34519618525593600

Offset: 0

Author

N. J. A. Sloane, Feb 01 2011

Keywords

• sign
• tabl

Comments

These are scaled central factorial numbers (see the discussion in the Comments section of A008955). The coefficients in the expansion of Product_{i=1..n} (x - i^2) give A008955, and the coefficients in the expansion of Product_{i=1..n} (x - (2i+1)^2) give A008956.

Examples

			Triangle begins:
 1
 1, -4
 1, -20, 64
 1, -56, 784, -2304
 1, -120, 4368, -52480, 147456
 1, -220, 16368, -489280, 5395456, -14745600
 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400
 1, -560, 119392, -12263680, 633721088, -15658639360, 157294854144, -416179814400
 1, -816, 262752, -42828032, 3773223168, -177891237888, 4165906530304, -40683662475264, 106542032486400
 1, -1140, 527136, -127959680, 17649505536, -1400415544320, 61802667606016, -1390437378293760, 13288048674471936, -34519618525593600
...
For example, for n=2, (x-4)(x-16) = x^2 - 20x + 64 => [1, -20, 64].
		

Links

• T. L. Curtright, D. B. Fairlie, and C. K. Zachos, A compact formula for rotations as spin matrix polynomials, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
• T. L. Curtright and T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arXiv:1408.0767 [math-ph], 2014.
• T. L. Curtright, More on Rotations as Spin Matrix Polynomials, arXiv preprint arXiv:1506.04648 [math-ph], 2015.

Crossrefs

Cf. A008955, A008956. This triangle is formed from the even-indexed rows of A182971 (the odd-indexed rows give A008956).

Cf. A160563.

Programs

Maple

Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
for n from 0 to 10 do
t1:=eval(Q(2*n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od:
# Using a bivariate generating function (adding a superdiagonal 1,0,0, ...):
gf := (t + sqrt(1 + t^2))^x:
ser := series(gf, t, 20): ct := n -> coeff(ser, t, n):
T := (n, k) -> n!*coeff(ct(n), x, n - k):
EvenPart := (T, len) -> local n, k;
seq(print(seq(T(n, k), k = 0..n, 2)), n = 0..2*len-1, 2):
EvenPart(T, 6);  # Peter Luschny, Mar 03 2024

Formula

Given a (0, 0)-based triangle U we call the triangle [U(n, k), k=0..n step 2, n=0..len step 2] the 'even subtriangle' of U. This triangle is the even subtriangle of U(n, k) = n! * [x^(n-k)] [t^n] (t + sqrt(1 + t^2))^x, albeit adding a superdiagonal 1, 0, 0, ... See A160563 for the odd subtriangle. - Peter Luschny, Mar 03 2024