A055451 Row sums of array in A055450.

1, 4, 13, 47, 173, 678, 2735, 11378, 48279, 208410, 911571, 4031919, 17999628, 81000573, 367040404, 1673295419, 7669312343, 35319197637, 163350479756, 758406642839, 3533447414030, 16514820417166, 77412170863861

Offset: 0

Author

Clark Kimberling, May 18 2000

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A055450, A055452, A055453, A055454, A055455.

Programs

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

Mathematica

T[n_, 0]:= 1; T[n_, k_]:= T[n, k]= If[1<=kA055451[n_]:= A055451[n]= Sum[T[n,k], {k,0,n}];
Table[A055451[n], {n,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 kA055451(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

Formula

a(n) = Sum_{k=0..n} A055450(n, k). - G. C. Greubel, Jan 29 2024