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A163840 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). 4
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; text; internal format)
OFFSET

0,2

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).

REFERENCES

Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.

LINKS

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

Peter Luschny, Swinging Factorial.

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

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);

MATHEMATICA

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 28 2013 *)

CROSSREFS

Row sums are A163843. Cf. A056040, A163865, A163841, A163842, A163650.

Sequence in context: A067323 A106534 A123346 * A122833 A193692 A075303

Adjacent sequences:  A163837 A163838 A163839 * A163841 A163842 A163843

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Aug 06 2009

STATUS

approved

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Last modified May 23 16:31 EDT 2017. Contains 286925 sequences.