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 A163840 Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040). 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. An analog to the binomial triangle of the factorials (A076571). REFERENCES Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008. LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened Peter Luschny, Swinging Factorial. FORMULA T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*i\$ where i\$ denotes the swinging factorial of i (A056040), for n >= 0, k >= 0. EXAMPLE Triangle begins     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-François 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,changed AUTHOR Peter Luschny, Aug 06 2009 STATUS approved

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