login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A189913 Triangle read by rows: T(n,k) = binomial(n, k) * k! / (floor(k/2)! * floor((k+2)/2)!). 1
1, 1, 1, 1, 2, 1, 1, 3, 3, 3, 1, 4, 6, 12, 2, 1, 5, 10, 30, 10, 10, 1, 6, 15, 60, 30, 60, 5, 1, 7, 21, 105, 70, 210, 35, 35, 1, 8, 28, 168, 140, 560, 140, 280, 14, 1, 9, 36, 252, 252, 1260, 420, 1260, 126, 126, 1, 10, 45, 360, 420, 2520, 1050, 4200, 630, 1260, 42 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The triangle may be regarded a generalization of the triangle A097610:

A097610(n,k) = binomial(n,k)*(2*k)$/(k+1);

T(n,k) = binomial(n,k)*(k)$/(floor(k/2)+1).

Here n$ denotes the swinging factorial A056040(n). As A097610 is a decomposition of the Motzkin numbers A001006, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.

T(n,n) = A057977(n) which can be seen as extended Catalan numbers.

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Peter Luschny, The lost Catalan numbers.

FORMULA

From R. J. Mathar, Jun 07 2011: (Start)

T(n,1) = n.

T(n,2) = A000217(n-1).

T(n,3) = A027480(n-2).

T(n,4) = A034827(n). (End)

EXAMPLE

[0]  1

[1]  1, 1

[2]  1, 2,  1

[3]  1, 3,  3,   3

[4]  1, 4,  6,  12,  2

[5]  1, 5, 10,  30, 10,  10

[6]  1, 6, 15,  60, 30,  60,  5

[7]  1, 7, 21, 105, 70, 210, 35, 35

MAPLE

A189913 := (n, k) -> binomial(n, k)*(k!/iquo(k, 2)!^2)/(iquo(k, 2)+1):

seq(print(seq(A189913(n, k), k=0..n)), n=0..7);

MATHEMATICA

T[n_, k_] := Binomial[n, k]*k!/((Floor[k/2])!*(Floor[(k + 2)/2])!); Table[T[n, k], {n, 0, 10}, {k, 0, n}]// Flatten (* G. C. Greubel, Jan 13 2018 *)

PROG

(PARI) {T(n, k) = binomial(n, k)*k!/((floor(k/2))!*(floor((k+2)/2))!) };

for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jan 13 2018

(MAGMA) /* As triangle */ [[Binomial(n, k)*Factorial(k)/(Factorial(Floor(k/2))*Factorial(Floor((k + 2)/2))): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 13 2018

CROSSREFS

Row sums are A189912.

Cf. A097610, A057977, A001006.

Sequence in context: A284834 A279677 A262180 * A240807 A283672 A053268

Adjacent sequences:  A189910 A189911 A189912 * A189914 A189915 A189916

KEYWORD

nonn,tabl,easy

AUTHOR

Peter Luschny, May 24 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 23 23:56 EST 2018. Contains 299595 sequences. (Running on oeis4.)