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%I #20 Jul 11 2019 14:31:17
%S 1,1,7,17,121,353,2591,8257,61921,207905,1582791,5501073,42344121,
%T 150827073,1170747519,4247388417,33186295681,122125206977,
%U 959260792775,3570473750929,28167068630713,105820555054241,837838806587167,3172136074486337
%N Schroeder triangle sums: a(2*n-1) = A010683(2*n-2) and a(2*n) = A010683(2*n-1) - A001003(2*n-1).
%C The terms of this sequence equal the Fi1 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows).
%F a(n) = Sum_{k=1..floor((n+1)/2)} A033877(2*k-1,n).
%F a(2*n-1) = A010683(2*n-2) and a(2*n) = A010683(2*n-1) - A001003(2*n-1).
%F G.f.: (1-4*x+x^2 - sqrt(1-6*x+x^2) + x*sqrt(1+6*x+x^2))/(8*x).
%p A227506 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if k<n then return(0) fi; T(n, k-1)+T(n-1, k-1)+T(n-1, k) end; add(T(2*k-1, n), k = 1..iquo(n+1,2)) end: seq(A227506(n), n = 1..24); # _Peter Luschny_, Jul 17 2013
%p A227506 := proc(n): if type(n, odd) then A010683(n-1) else A010683(n-1) - A001003(n-1) fi: end: A010683 := proc(n): if n = 0 then 1 else (2/n)*add(binomial(n, k)* binomial(n+k+1, k-1), k=1..n) fi: end: A001003 := proc(n): if n = 0 then 1 else add(binomial(n, j)*binomial(n+j, n-1), j=0..n)/(2*n) fi: end: seq(A227506(n), n=1..24);
%t T[n_, k_] := T[n, k] = Which[n == 1, 1, k < n, 0, True, T[n, k - 1] + T[n - 1, k - 1] + T[n - 1, k]];
%t a[n_] := Sum[T[2 k - 1, n], {k, 1, (n + 1)/2}];
%t Array[a, 24] (* _Jean-François Alcover_, Jul 11 2019, from Sage *)
%o (Sage)
%o def A227506(n):
%o @CachedFunction
%o def T(n, k):
%o if n==1: return 1
%o if k <n: return 0
%o return T(n,k-1)+T(n-1,k-1)+T(n-1,k)
%o return add(T(2*k-1,n) for k in (1..(n+1)//2))
%o [A227506(n) for n in (1..24)] # _Peter Luschny_, Jul 16 2013
%Y Cf. A033877, A010683, A001003.
%K nonn,easy
%O 1,3
%A _Johannes W. Meijer_, Jul 15 2013
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