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A023660 Convolution of odd numbers and A023533.

%I A023660 #7 Jul 18 2022 09:59:53
%S A023660 1,3,5,8,12,16,20,24,28,33,39,45,51,57,63,69,75,81,87,94,102,110,118,
%T A023660 126,134,142,150,158,166,174,182,190,198,206,215,225,235,245,255,265,
%U A023660 275,285,295,305,315,325,335,345,355,365,375,385,395,405
%N A023660 Convolution of odd numbers and A023533.
%H A023660 G. C. Greubel, <a href="/A023660/b023660.txt">Table of n, a(n) for n = 1..5000</a>
%F A023660 From _G. C. Greubel_, Jul 17 2022: (Start)
%F A023660 a(n) = Sum_{j=0..n-1} (2*j+1)*A023533(n-j).
%F A023660 a(n) = 2*A023543(n-1) + A056556(n).
%F A023660 T(n, k) = (2*k+1)*n + 6*binomial(n+2, 4), for 0 <= k <= n*(n+3)/2 and n >= 1 (as an irregular triangle). (End)
%t A023660 Table[(2*k+1)*n + 6*Binomial[n+2,4], {n, 7}, {k,0,n*(n+3)/2}]//Flatten (* _G. C. Greubel_, Jul 17 2022 *)
%o A023660 (Magma)
%o A023660 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A023660 [(&+[(2*k+1)*A023533(n-k): k in [0..n-1]]): n in [1..80]]; // _G. C. Greubel_, Jul 17 2022
%o A023660 (SageMath)
%o A023660 def A023660(n, k): return (2*k+1)*n + 6*binomial(n+2, 4)
%o A023660 flatten([[A023660(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # _G. C. Greubel_, Jul 17 2022
%Y A023660 Cf. A005408, A023533, A023543, A056556.
%K A023660 nonn
%O A023660 1,2
%A A023660 _Clark Kimberling_