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

0,2

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]]];
A055451[n_]:= A055451[n]= Sum[T[n, k], {k, 0, n}];
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

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

Sequence in context: A363547 A017944 A017945 * A163862 A149440 A149441

Adjacent sequences: A055448 A055449 A055450 * A055452 A055453 A055454

Keyword

nonn

Author

Clark Kimberling, May 18 2000

Status

approved