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

0,5

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.)

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Peter Luschny, The lost Catalan numbers.

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).

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-François Alcover, Jul 29 2013 *)

Prog

(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

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