OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
O.g.f.: F(x^2)*(1/(1-x*S(x^2))+C(x^2)*x/(1-x*C(x^2))), where C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108, S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318, and F(x)=S(x)/(1-x*C(x)*S(x)) is o.g.f. for A026781. - Max Alekseyev, Jan 13 2015
C(x^2)/(1-x*C(x^2)) above is the o.g.f. for A001405. 1/(1-x*S(x^2)) above is the o.g.f for A026003 starting with an additional 1: 1,1,1,3,5,13,25,... - R. J. Mathar, Feb 10 2022
MAPLE
T:= proc(n, k) option remember;
if n<0 then 0;
elif k=0 or k =n then 1;
elif k <= n/2 then
procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
else
procname(n-1, k-1)+procname(n-1, k) ;
fi ;
end proc:
seq( add(T(n, k), k=0..n), n=0..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[Sum[T[n, k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Nov 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n<0): return 0
elif (k==0 or k==n): return 1
elif (k<=n/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n, k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 02 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Max Alekseyev, Jan 13 2015
STATUS
approved