A027219 a(n) = Sum_{k=0..n} (k+1) * A026736(n,k).

1, 3, 8, 20, 50, 117, 283, 639, 1512, 3338, 7774, 16898, 38884, 83566, 190488, 405848, 918120, 1942813, 4367665, 9191499, 20555546, 43061789, 95874233, 200083005, 443770612, 923124007, 2040635445, 4233080627, 9330343290

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A026736.

Programs

GAP

T:= function(n, k)
    if k=0 or k=n then return 1;
    elif k=n-1 then return n;
    elif (n mod 2)=0 and k=Int((n-2)/2) then 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);
    fi;
  end;
List([0..20], n-> Sum([0..n], k-> (k+1)*T(n, k) )); # G. C. Greubel, Jul 19 2019

Mathematica

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/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[(k+1)*T[n,k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Jul 19 2019 *)

PARI

T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(20, n, n--; sum(k=0, n, (k+1)*T(n, k)) ) \\ G. C. Greubel, Jul 19 2019

Sage

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n, 2)==0 and k==(n-2)/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((k+1)*T(n,k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jul 19 2019