A026779 - OEIS

%I #8 Nov 01 2019 22:23:41

%S 1,1,2,3,6,10,17,32,56,97,181,322,567,1053,1892,3369,6241,11286,20255,

%T 37463,68044,122809,226896,413376,749159,1382990,2525162,4590351,

%U 8468738,15487526,28218889,52035094,95273724,173898941

%N a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026769.

%H G. C. Greubel, <a href="/A026779/b026779.txt">Table of n, a(n) for n = 0..1000</a>

%p T:= proc(n,k) option remember;

%p    if n<0 then 0;

%p    elif k=0 or k=n then 1;

%p    elif n=2 and k=1 then 2;

%p    elif k <= (n-1)/2 then

%p         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;

%p    else

%p        procname(n-1,k-1)+procname(n-1,k) ;

%p    end if ;

%p end proc;

%p seq(add(T(n-k,k), k=0..floor(n/2)), n=0..30); # _G. C. Greubel_, Nov 01 2019

%t T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/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], {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Nov 01 2019 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o     if (k==0 or k==n): return 1

%o     elif (n==2 and k==1): return 2

%o     elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)

%o     else: return T(n-1,k-1) + T(n-1,k)

%o [sum(T(n-k,k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Nov 01 2019

%Y Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026778.

%K nonn

%O 0,3

%A _Clark Kimberling_