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

%I A182867 #40 Mar 03 2024 14:13:28
%S A182867 1,1,-4,1,-20,64,1,-56,784,-2304,1,-120,4368,-52480,147456,1,-220,
%T A182867 16368,-489280,5395456,-14745600,1,-364,48048,-2846272,75851776,
%U A182867 -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
%N A182867 Triangle read by rows: row n gives coefficients in expansion of Product_{i=1..n} (x - (2i)^2), highest powers first.
%C A182867 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.
%H A182867 T. L. Curtright, D. B. Fairlie, and C. K. Zachos, <a href="http://arxiv.org/abs/1402.3541">A compact formula for rotations as spin matrix polynomials</a>, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
%H A182867 T. L. Curtright and T. S. Van Kortryk, <a href="http://arxiv.org/abs/1408.0767">On Rotations as Spin Matrix Polynomials</a>, arXiv:1408.0767 [math-ph], 2014.
%H A182867 T. L. Curtright, <a href="http://arxiv.org/abs/1506.04648">More on Rotations as Spin Matrix Polynomials</a>, arXiv preprint arXiv:1506.04648 [math-ph], 2015.
%F A182867 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
%e A182867 Triangle begins:
%e A182867  1
%e A182867  1, -4
%e A182867  1, -20, 64
%e A182867  1, -56, 784, -2304
%e A182867  1, -120, 4368, -52480, 147456
%e A182867  1, -220, 16368, -489280, 5395456, -14745600
%e A182867  1, -364, 48048, -2846272, 75851776, -791691264, 2123366400
%e A182867  1, -560, 119392, -12263680, 633721088, -15658639360, 157294854144, -416179814400
%e A182867  1, -816, 262752, -42828032, 3773223168, -177891237888, 4165906530304, -40683662475264, 106542032486400
%e A182867  1, -1140, 527136, -127959680, 17649505536, -1400415544320, 61802667606016, -1390437378293760, 13288048674471936, -34519618525593600
%e A182867 ...
%e A182867 For example, for n=2, (x-4)(x-16) = x^2 - 20x + 64 => [1, -20, 64].
%p A182867 Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
%p A182867 else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
%p A182867 for n from 0 to 10 do
%p A182867 t1:=eval(Q(2*n)); t1d:=degree(t1);
%p A182867 t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
%p A182867 lprint(t2);
%p A182867 od:
%p A182867 # Using a bivariate generating function (adding a superdiagonal 1,0,0, ...):
%p A182867 gf := (t + sqrt(1 + t^2))^x:
%p A182867 ser := series(gf, t, 20): ct := n -> coeff(ser, t, n):
%p A182867 T := (n, k) -> n!*coeff(ct(n), x, n - k):
%p A182867 EvenPart := (T, len) -> local n, k;
%p A182867 seq(print(seq(T(n, k), k = 0..n, 2)), n = 0..2*len-1, 2):
%p A182867 EvenPart(T, 6);  # _Peter Luschny_, Mar 03 2024
%Y A182867 Cf. A008955, A008956. This triangle is formed from the even-indexed rows of A182971 (the odd-indexed rows give A008956).
%Y A182867 Cf. A160563.
%K A182867 sign,tabl
%O A182867 0,3
%A A182867 _N. J. A. Sloane_, Feb 01 2011