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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A163649 Triangle interpolating between (-1)^n (A033999) and A056040(n), read by rows. 3
1, -1, 1, 1, -2, 2, -1, 3, -6, 6, 1, -4, 12, -24, 6, -1, 5, -20, 60, -30, 30, 1, -6, 30, -120, 90, -180, 20, -1, 7, -42, 210, -210, 630, -140, 140, 1, -8, 56, -336, 420, -1680, 560, -1120, 70 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

T(n,k) = (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)!

Let A(n,k) = abs(T(n,k)) be the coefficients of the polynomials Sum_{k=0..n} binomial(n,k)*A056040(k)*q^k. Substituting q^k -> 1/(floor(k/2)+1) in the polynomials gives the extended Motzkin numbers A189912. (See A089627 for the Motzkin numbers and A194586 for the complementary Motzkin numbers.)

LINKS

Table of n, a(n) for n=0..44.

Peter Luschny, The lost Catalan numbers.

FORMULA

egf(x,y) = exp(-x)*BesselI(0,2*x*y)*(1+x*y).

EXAMPLE

1

-1, 1

1, -2, 2

-1, 3, -6, 6

1, -4, 12, -24, 6

-1, 5, -20, 60, -30, 30

1, -6, 30, -120, 90, -180, 20

-1, 7, -42, 210, -210, 630, -140, 140

1, -8, 56, -336, 420, -1680, 560, -1120, 70

MAPLE

a := proc(n, k) (-1)^(n-k)*floor(k/2)!^(-2)*n!/(n-k)! end:

seq(print(seq(a(n, k), k=0..n)), n=0..8);

MATHEMATICA

t[n_, k_] := (-1)^(n - k)*Floor[k/2]!^(-2)*n!/(n - k)!; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 29 2013 *)

CROSSREFS

Row sums give A163650, row sums of absolute values give A163865.

Aerated versions A194586 (odd case) and A089627 (even case).

Sequence in context: A247507 A107111 A082037 * A110858 A008279 A239572

Adjacent sequences:  A163646 A163647 A163648 * A163650 A163651 A163652

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Aug 02 2009

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 March 27 06:50 EDT 2017. Contains 284144 sequences.