A026787 a(n) = Sum_{k=0..n} T(n,k), T given by A026780.

1, 2, 5, 11, 26, 58, 136, 306, 717, 1625, 3813, 8697, 20451, 46909, 110563, 254855, 602042, 1393746, 3299304, 7666786, 18182976, 42391546, 100704606, 235452416, 560147414, 1312916040, 3127406812, 7346213746, 17518138314, 41228281888, 98408997716, 231990850378, 554207752781, 1308436686305, 3128033585157

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026788, A026789, A026790.

Programs

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 *)

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

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

Extensions

More terms from Max Alekseyev, Jan 13 2015