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 A163842 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial transform. Same as interpolating the beta numbers 1/beta(n,n) (A002457) with (A163869). 5
 1, 7, 6, 43, 36, 30, 249, 206, 170, 140, 1395, 1146, 940, 770, 630, 7653, 6258, 5112, 4172, 3402, 2772, 41381, 33728, 27470, 22358, 18186, 14784, 12012, 221399, 180018, 146290, 118820, 96462, 78276, 63492 (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)*(2i+1)\$ 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 G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened Peter Luschny, Swinging Factorial. EXAMPLE 1 7, 6 43, 36, 30 249, 206, 170, 140 1395, 1146, 940, 770, 630 7653, 6258, 5112, 4172, 3402, 2772 41381, 33728, 27470, 22358, 18186, 14784, 12012 MAPLE Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840. a := n -> SumTria(k->swing(2*k+1), n, true); MATHEMATICA sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n-k, n-i]*sf[2*i+1], {i, k, n}]; Table[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *) CROSSREFS Row sums are A163845. Cf. A056040, A163650, A163841, A163842, A163840, A002426, A000984. Sequence in context: A163260 A073112 A070425 * A288245 A038272 A249992 Adjacent sequences:  A163839 A163840 A163841 * A163843 A163844 A163845 KEYWORD nonn,tabl AUTHOR Peter Luschny, Aug 06 2009 STATUS approved

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