login
A transform of the Motzkin numbers.
2

%I #9 Feb 09 2015 13:59:42

%S 1,-1,-1,0,2,1,-1,-4,-2,5,11,0,-21,-24,18,68,42,-100,-203,-17,428,521,

%T -340,-1544,-1019,2252,4892,606,-10297,-13331,7821,39310,28028,-56893,

%U -130394,-22239,272991,370641,-193874,-1081694,-821669,1536026,3707766,798376

%N A transform of the Motzkin numbers.

%C Hankel transform is A157144.

%F G.f.: (1-x)/(1+x^2+x^3)*c((x/(1+x^2+x^3))^2), c(x) the g.f. of A000108. - (The formula does not match the entries, even if A000108 is replaced by A001006. _R. J. Mathar_, Feb 06 2015)

%F a(n)=sum{k=0..n, (-1)^C(n-k+1,2)*C(floor((n-k)/2),k)*A001006(k)}.

%o (PARI) A157143(n)=vector(n++,k,(-1)^binomial(n+1-k,2)*binomial((n-k)\2,k-1))*Vec(serreverse(x/(1+x+x^2+O(x^n))))~ \\ _M. F. Hasler_, Feb 09 2015

%Y Cf. A001006.

%K easy,sign

%O 0,5

%A _Paul Barry_, Feb 24 2009