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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

LINKS

Table of n, a(n) for n=0..54.

T. L. Curtright, D. B. Fairlie, C. K. Zachos, A compact formula for rotations as spin matrix polynomials, arXiv preprint arXiv:1402.3541, 2014

T. L. Curtright, T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arXiv:1408.0767, 2014.

T. L. Curtright, More on Rotations as Spin Matrix Polynomials, arXiv preprint arXiv:1506.04648, 2015

EXAMPLE

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].

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:

CROSSREFS

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

Sequence in context: A144354 A049352 A322218 * A182826 A144484 A121336

Adjacent sequences:  A182864 A182865 A182866 * A182868 A182869 A182870

KEYWORD

sign,tabl

AUTHOR

N. J. A. Sloane, Feb 01 2011

STATUS

approved

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Last modified September 26 20:34 EDT 2021. Contains 347672 sequences. (Running on oeis4.)