A163770 -id:A163770

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A163773

Row sums of the swinging derangement triangle (A163770).

+20
2

1, 1, 4, 15, -14, 185, -454, 2107, -6194, 22689, -70058, 234971, -734304, 2368379, -7404318, 23417955, -72988938, 228324569, -708982738, 2202742447, -6815736144, 21077285943, -65016664062, 200371842727, -616463969324, 1894794918275, -5816606133674, 17839764136377

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Peter Luschny, Swinging Factorial.

FORMULA

a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040).

MAPLE

swing := proc(n) option remember; if n = 0 then 1 elif

irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:

a := proc(n) local i, k; add(add((-1)^(n-i)*binomial(n-k, n-i)*swing(i), i=k..n), k=0..n) end:

MATHEMATICA

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 03 2017 *)

CROSSREFS

Cf. A163770.

KEYWORD

sign

AUTHOR

Peter Luschny, Aug 05 2009

EXTENSIONS

Terms a(18) onward added by G. C. Greubel, Aug 03 2017

STATUS

approved

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).

+10
6

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).

This is also the square array of central binomial coefficients A000984 in column 0 and higher (first: A051924, second, etc.) differences in subsequent columns, read by antidiagonals. - M. F. Hasler, Nov 15 2019

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

Triangle begins

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;

From M. F. Hasler, Nov 15 2019: (Start)

The square array having central binomial coefficients A000984 in column 0 and higher differences in subsequent columns (col. 1 = A051924) starts:

1 1 3 7 19 51 ...

2 4 10 26 70 192 ...

6 14 36 96 262 726 ...

20 50 132 358 988 2760 ...

70 182 490 1346 3748 10540 ...

252 672 1836 5094 14288 40404 ...

(...)

Read by falling antidiagonals this yields the same sequence. (End)

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-François Alcover, Jun 28 2013 *)

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Aug 05 2009

STATUS

approved

A163772

Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse.

+10
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

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.

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Aug 05 2009

STATUS

approved

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