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A163840
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Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040). An analog to the binomial triangle of the factorials (A076571).
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4
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1, 2, 1, 5, 3, 2, 16, 11, 8, 6, 47, 31, 20, 12, 6, 146, 99, 68, 48, 36, 30, 447, 301, 202, 134, 86, 50, 20, 1380, 933, 632, 430, 296, 210, 160, 140, 4251, 2871, 1938, 1306, 876, 580, 370, 210, 70, 13102, 8851, 5980, 4042, 2736, 1860, 1280, 910, 700, 630
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Triangle read by rows. For n >= 0, k >= 0 let
T(n,k) = sum{i=k..n} binomial(n-k,n-i)*i$
where i$ denotes the swinging factorial of i (A056040).
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REFERENCES
| Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.
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LINKS
| Peter Luschny, Swinging Factorial.
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EXAMPLE
| 1
2, 1
5, 3, 2
16, 11, 8, 6
47, 31, 20, 12, 6
146, 99, 68, 48, 36, 30
447, 301, 202, 134, 86, 50, 20
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MAPLE
| SumTria := proc(f, n, display) local m, A, j, i, T; T:=f(0);
for m from 0 by 1 to n-1 do A[m] := f(m);
for j from m by -1 to 1 do A[j-1] := A[j-1] + A[j] od;
for i from 0 to m do T := T, A[i] od;
if display then print(seq(T[i], i=nops([T])-m..nops([T]))) fi;
od; subsop(1=NULL, [T]) end:
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
Computes n rows of the triangle:
A163840 := n -> SumTria(swing, n, true);
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CROSSREFS
| Sum rows are A163843. Cf. A056040, A163865, A163841, A163842, 163650.
Sequence in context: A067323 A106534 A123346 * A122833 A193692 A075303
Adjacent sequences: A163837 A163838 A163839 * A163841 A163842 A163843
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KEYWORD
| nonn,tabl
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AUTHOR
| Peter Luschny (peter(AT)luschny.de), Aug 06 2009
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