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A288580
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Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-k*i| <= n} (n-k*i), with n >= 0, k >= 1.
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10
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1, 1, -1, 1, -1, 4, 1, 1, -4, -36, 1, 1, -2, 9, 576, 1, 1, -4, -9, 64, -14400, 1, 1, 2, -3, -8, -225, 518400, 1, 1, 2, -6, -16, 40, -2304, -25401600, 1, 1, 2, -9, -4, -15, 324, 11025, 1625702400, 1, 1, 2, 3, -8, -25, 144, 280, 147456, -131681894400, 1, 1, 2, 3, -12, -5, -24, 105, -2240, -893025, 13168189440000
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,6
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REFERENCES
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F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
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LINKS
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Table of n, a(n) for n=0..65.
J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
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EXAMPLE
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Array begins:
1, -1, 4, -36, 576, -14400, 518400, -25401600, 1625702400, -131681894400, ...
1, -1, -4, 9, 64, -225, -2304, 11025, 147456, -893025, -14745600, 108056025, ...
1, 1, -2, -9, -8, 40, 324, 280, -2240, -26244, -22400, 246400, 3779136, ...
1, 1, -4, -3, -16, -15, 144, 105, 1024, 945, -14400, -10395, -147456, ...
1, 1, 2, -6, -4, -25, -24, -42, 336, 216, 2500, 2376, 4032, ...
1, 1, 2, -9, -8, -5, -36, -35, -64, 729, 640, 385, 5184, ...
1, 1, 2, 3, -12, -10, -6, -49, -48, -90, -120, 1320, 1080, ...
1, 1, 2, 3, -16, -15, -12, -7, -64, -63, -120, -165, 2304, ...
1, 1, 2, 3, 4, -20, -18, -14, -8, -81, -80, -154, -216, ...
1, 1, 2, 3, 4, -25, -24, -21, -16, -9, -100, -99, -192, ...
...
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MAPLE
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T:=proc(n, k) local i, p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
scan1:=proc(a, M1) local lis, n, k; lis:=[]; for n from 1 to M1 do for k from 0 to n-1 do
lis:=[op(lis), a(k, n-k)]; od: od: lis; end:
scan1(T, 12);
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MATHEMATICA
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T[n_, k_] := Module[{i, p = 1}, For[i = 0, i <= Floor[2n/k], i++, If[n - k i != 0, p *= (n - k i)]]; p]; T[_, 0] = 1;
Table[T[k, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 05 2020, after Maple *)
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CROSSREFS
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Rows k=1 through 9 are signed A001044 or A092396, signed A184877 or A092397, A092398, A092399, A092971, A092972, A092973, A092974,
Sequence in context: A350000 A193752 A136680 * A035589 A021247 A016522
Adjacent sequences: A288577 A288578 A288579 * A288581 A288582 A288583
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KEYWORD
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sign,tabl
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AUTHOR
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N. J. A. Sloane, Jul 03 2017
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STATUS
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approved
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