A163649 Triangle interpolating between (-1)^n (A033999) and A056040(n), read by rows.

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

Offset: 0

Author

Peter Luschny, Aug 02 2009

Keywords

• sign
• tabl

Comments

Given 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.)

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Peter Luschny, The lost Catalan numbers.

Crossrefs

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

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

Programs

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-François Alcover, Jul 29 2013 *)

PARI

for(n=0,10, for(k=0,n, print1((-1)^(n -k)*( (floor(k/2))! )^(-2)*(n!/(n - k)!), ", "))) \\ G. C. Greubel, Aug 01 2017

Formula

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

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