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 A163772 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse. 5
 1, 5, 6, 19, 24, 30, 67, 86, 110, 140, 227, 294, 380, 490, 630, 751, 978, 1272, 1652, 2142, 2772, 2445, 3196, 4174, 5446, 7098, 9240, 12012, 7869, 10314, 13510, 17684, 23130, 30228, 39468, 51480 (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} (-1)^(n-i)*binomial(n-k,n-i)*(2i+1)\$  where i\$ denotes the swinging factorial of i (A056040). LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011. Peter Luschny, Swinging Factorial. M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1. EXAMPLE 1 5, 6 19, 24, 30 67, 86, 110, 140 227, 294, 380, 490, 630 751, 978, 1272, 1652, 2142, 2772 2445, 3196, 4174, 5446, 7098, 9240, 12012 MAPLE For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle. a := n -> DiffTria(k->swing(2*k+1), n, true); MATHEMATICA sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[ (-1)^(n-i)*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 A163775. Cf. A056040, A163650, A163771, A163772, A002426, A000984. Sequence in context: A240400 A031448 A290254 * A056509 A129722 A133608 Adjacent sequences:  A163769 A163770 A163771 * A163773 A163774 A163775 KEYWORD nonn,tabl AUTHOR Peter Luschny, Aug 05 2009 STATUS approved

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Last modified January 27 18:57 EST 2021. Contains 340479 sequences. (Running on oeis4.)