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 A163841 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths (A026375) with the central binomial coefficients (A000984). 4
 1, 3, 2, 11, 8, 6, 45, 34, 26, 20, 195, 150, 116, 90, 70, 873, 678, 528, 412, 322, 252, 3989, 3116, 2438, 1910, 1498, 1176, 924, 18483, 14494, 11378, 8940, 7030, 5532, 4356, 3432, 86515, 68032, 53538 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For n >= 0, k >= 0 let T(n,k) = sum{i=k..n} binomial(n-k,n-i)*(2i)\$ where i\$ denotes the swinging factorial of i (A056040). Triangle read by rows. 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. Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7. EXAMPLE Triangle begins      1;      3,    2;     11,    8,    6;     45,   34,   26,   20;    195,  150,  116,   90,   70;    873,  678,  528,  412,  322,  252;   3989, 3116, 2438, 1910, 1498, 1176,  924; MAPLE Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840. a := n -> SumTria(k->swing(2*k), n, true); MATHEMATICA sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *) CROSSREFS Row sums are A163844. Cf. A056040, A163650, A163841, A163842, A163840, A026375, A002426, A000984. Sequence in context: A013945 A072656 A191669 * A276589 A275950 A276587 Adjacent sequences:  A163838 A163839 A163840 * A163842 A163843 A163844 KEYWORD nonn,tabl AUTHOR Peter Luschny, Aug 06 2009 STATUS approved

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