OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
FORMULA
a(n) = Sum_{k=0..n} A055450(n, k). - G. C. Greubel, Jan 29 2024
MATHEMATICA
T[n_, 0]:= 1; T[n_, k_]:= T[n, k]= If[1<=k<n/2, T[n-1, k-1] + T[n-1, k], If[k==n/2, T[n-2, k-1] + T[n-1, k-1], T[n+1, k] + T[n-1, k-1]]];
Table[A055451[n], {n, 0, 40}] (* G. C. Greubel, Jan 29 2024 *)
PROG
(Magma)
B:=Binomial; G:=Gamma; F:=Factorial;
p:= func< n, k, j | B(n-2*k+j-1, j)*G(n-k+j+3/2)/(F(j)*G(n-k+3/2)*B(n-k+j+2, j)) >;
f:= func< n, k | (n-k+1)*Binomial(n+k, k)/(n+1) >;
function T(n, k) // T = A055450
if k lt n/2 then return f(n-k+1, k);
else return Round(Catalan(n-k+1)*(&+[p(n, k, j)*(-4)^j: j in [0..n]]));
end if;
end function;
A055451:= func< n | (&+[T(n, k): k in [0..n]]) >;
[A055451(n): n in [0..40]]; // G. C. Greubel, Jan 29 2024
(SageMath)
def f(n, k): return (n-k+1)*binomial(n+k, k)/(n+1)
def T(n, k): # T = A055450
if k<n/2: return f(n-k+1, k)
else: return round(catalan_number(n-k+1)*hypergeometric([n-2*k, (3+2*(n-k))/2], [3+n-k], -4))
def A055451(n): return sum(T(n, k) for k in range(n+1))
[A055451(n) for n in range(41)] # G. C. Greubel, Jan 30 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 18 2000
STATUS
approved