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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

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proposed

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proposed

Ta(n,) = Sum_{k=0) + T(..n,1) + ... } T(n,nk), T given by A026780.

G. C. Greubel, <a href="/A026787/b026787.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

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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

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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

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 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

More terms from Max Alekseyev, Jan 13 2015

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