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A055451 Row sums of array in A055450.

%I A055451 #14 Jan 31 2024 07:58:21
%S A055451 1,4,13,47,173,678,2735,11378,48279,208410,911571,4031919,17999628,
%T A055451 81000573,367040404,1673295419,7669312343,35319197637,163350479756,
%U A055451 758406642839,3533447414030,16514820417166,77412170863861
%N A055451 Row sums of array in A055450.
%H A055451 G. C. Greubel, <a href="/A055451/b055451.txt">Table of n, a(n) for n = 0..250</a>
%F A055451 a(n) = Sum_{k=0..n} A055450(n, k). - _G. C. Greubel_, Jan 29 2024
%t A055451 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]]];
%t A055451 A055451[n_]:= A055451[n]= Sum[T[n,k], {k,0,n}];
%t A055451 Table[A055451[n], {n,0,40}] (* _G. C. Greubel_, Jan 29 2024 *)
%o A055451 (Magma)
%o A055451 B:=Binomial; G:=Gamma; F:=Factorial;
%o A055451 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)) >;
%o A055451 f:= func< n,k | (n-k+1)*Binomial(n+k, k)/(n+1) >;
%o A055451 function T(n,k) // T = A055450
%o A055451   if k lt n/2 then return f(n-k+1, k);
%o A055451   else return Round(Catalan(n-k+1)*(&+[p(n,k,j)*(-4)^j: j in [0..n]]));
%o A055451   end if;
%o A055451 end function;
%o A055451 A055451:= func< n | (&+[T(n,k): k in [0..n]]) >;
%o A055451 [A055451(n): n in [0..40]]; // _G. C. Greubel_, Jan 29 2024
%o A055451 (SageMath)
%o A055451 def f(n,k): return (n-k+1)*binomial(n+k, k)/(n+1)
%o A055451 def T(n,k): # T = A055450
%o A055451     if k<n/2: return f(n-k+1,k)
%o A055451     else: return round(catalan_number(n-k+1)*hypergeometric([n-2*k, (3+2*(n-k))/2], [3+n-k], -4))
%o A055451 def A055451(n): return sum(T(n,k) for k in range(n+1))
%o A055451 [A055451(n) for n in range(41)] # _G. C. Greubel_, Jan 30 2024
%Y A055451 Cf. A055450, A055452, A055453, A055454, A055455.
%K A055451 nonn
%O A055451 0,2
%A A055451 _Clark Kimberling_, May 18 2000