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A163771 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984). 4
1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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)$ 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

Triangle begins

    1;

    1,   2;

    3,   4,   6;

    7,  10,  14,  20;

   19,  26,  36,  50,  70;

   51,  70,  96, 132, 182, 252;

  141, 192, 262, 358, 490, 672, 924;

MAPLE

For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.

a := n -> DiffTria(k->swing(2*k), 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], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 28 2013 *)

CROSSREFS

Row sums are A163774. Cf. A056040, A163650, A163771, A163772, A002426, A000984.

Sequence in context: A237752 A032480 A226137 * A194855 A272766 A271340

Adjacent sequences:  A163768 A163769 A163770 * A163772 A163773 A163774

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Aug 05 2009

STATUS

approved

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Last modified August 20 17:05 EDT 2017. Contains 290836 sequences.