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A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)*(x-(n-2)^2)*(x-(n-4)^2)*...*(x-(1 or 2)^2), highest powers first. 5
1, 1, -1, 1, -4, 1, -10, 9, 1, -20, 64, 1, -35, 259, -225, 1, -56, 784, -2304, 1, -84, 1974, -12916, 11025, 1, -120, 4368, -52480, 147456, 1, -165, 8778, -172810, 1057221, -893025, 1, -220, 16368, -489280, 5395456, -14745600, 1, -286, 28743, -1234948, 21967231, -128816766, 108056025, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

These are scaled versions of the central factorial numbers in A008955 and A008956.

Even-indexed rows give A182867, odd-indexed rows give A008956.

A121408 is an unsigned and aerated version of the row reverse of this triangle. - Peter Bala, Aug 29 2012

LINKS

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

FORMULA

For n even, let Q(n) = Product_{i=1..n/2} (x - (2*i)^2) and for n odd let Q(n) = Product_{i=0..(n-1)/2} (x - (2i+1)^2). n-th row of triangle gives coefficients in expansion of Q(n).

EXAMPLE

Triangle begins:

1

1, -1

1, -4

1, -10, 9

1, -20, 64

1, -35, 259, -225

1, -56, 784, -2304

1, -84, 1974, -12916, 11025

1, -120, 4368, -52480, 147456

1, -165, 8778, -172810, 1057221, -893025

1, -220, 16368, -489280, 5395456, -14745600

...

E.g. for n=5 Q(5) = (x-1^2)*(x-3^2)*(x-5^2) = x^3-35*x^2+259*x-225.

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 12 do

t1:=eval(Q(n)); t1d:=degree(t1);

t12:=y^t1d*subs(x=1/y, t1); t2:=seriestolist(series(t12, y, 20));

lprint(t2);

od:

CROSSREFS

Even-indexed rows give A182867, odd-indexed rows give A008956.

Column 1,4,10,20, ... is A000292. The next two columns give A181888, A184878. The last diagonal is A184877.

Cf. A008955, A008956. A121408.

Sequence in context: A185676 A277583 A213765 * A062145 A178216 A307529

Adjacent sequences:  A182968 A182969 A182970 * A182972 A182973 A182974

KEYWORD

sign,tabf

AUTHOR

N. J. A. Sloane, Feb 01 2011

STATUS

approved

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Last modified February 24 20:30 EST 2020. Contains 332210 sequences. (Running on oeis4.)