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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).

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.

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-Fran├žois Alcover, Jun 28 2013 *)

CROSSREFS

Row sums are A163775. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

Sequence in context: A257331 A240400 A031448 * 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 April 25 19:05 EDT 2017. Contains 285426 sequences.