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a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n-k+1), where k = floor(n/2), s = A023533.
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%I #11 Jan 09 2023 07:40:57

%S 0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,

%T 0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,

%U 0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0

%N a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n-k+1), where k = floor(n/2), s = A023533.

%H G. C. Greubel, <a href="/A025125/b025125.txt">Table of n, a(n) for n = 1..5000</a>

%t b[j_]:= b[j]= Sum[KroneckerDelta[j, Binomial[m+2,3]], {m,0,15}];

%t A025125[n_]:= A025125[n]= Sum[b[n -j+1]*b[j+1], {j, Floor[(n+2)/2], n}];

%t Table[A025125[n], {n,130}] (* _G. C. Greubel_, Sep 14 2022 *)

%o (Magma)

%o A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;

%o A025125:= func< n | (&+[A023533(k)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;

%o [A025125(n): n in [1..130]]; // _G. C. Greubel_, Sep 14 2022

%o (SageMath)

%o @CachedFunction

%o def b(j): return sum(bool(j==binomial(m+2,3)) for m in (0..15))

%o @CachedFunction

%o def A025125(n): return sum(b(n-j+1)*b(j+1) for j in (((n+2)//2)..n))

%o [A025125(n) for n in (1..130)] # _G. C. Greubel_, Sep 14 2022

%Y Cf. A023533.

%K nonn

%O 1,138

%A _Clark Kimberling_