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%I #10 Jul 27 2023 08:29:25
%S 1,1,3,13,44,146,530,1975,7314,27262,102802,390138,1486064,5682756,
%T 21812436,83976075,324115550,1253795510,4859960402,18871869302,
%U 73398851448,285882923196,1114943553308,4353426835238,17016813133124,66581653586476,260750813149140,1022023318047220
%N a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).
%C Narayana transform of A027656.
%H G. C. Greubel, <a href="/A136520/b136520.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).
%F a(n) = Sum_{j=0..floor((n-1)/2)} ((j+1)/(2*j+1))*binomial(n, 2*j) * binomial(n-1, 2*j). - _G. C. Greubel_, Jul 27 2023
%e a(4) = 13 = (1, 6, 6, 1) dot (1, 0, 2, 0) = (1 + 0 + 12 + 0).
%e Triangle A001263(n,k) * A027656(k+1) and the rows sums:
%e 1; : 1;
%e 1, 0; : 1;
%e 1, 0, 2; : 3;
%e 1, 0, 12, 0; : 13;
%e 1, 0, 40, 0, 3; : 44;
%e 1, 0, 100, 0, 45, 0; : 146;
%e 1, 0, 210, 0, 315, 0, 4; : 530;
%e 1, 0, 392, 0, 1470, 0, 112, 0; : 1975;
%e 1, 0, 672, 0, 5292, 0, 1344, 0, 5; : 7314;
%e 1, 0, 1080, 0, 15876, 0, 10080, 0, 225, 0 : 27262;
%t A136520[n_]:= Sum[Binomial[n-1, 2*k]*Binomial[n, 2*k]*((k+1)/(2*k+1)), {k,0,Floor[(n-1)/2]}];
%t Table[A136520[n], {n, 40}] (* _G. C. Greubel_, Jul 27 2023 *)
%o (Magma)
%o A136520:= func< n | (&+[((j+1)/(2*j+1))*Binomial(n,2*j)*Binomial(n-1,2*j): j in [0..Floor((n-1)/2)]]) >;
%o [A136520(n): n in [1..40]]; // _G. C. Greubel_, Jul 27 2023
%o (SageMath)
%o def A136520(n): return sum(((j+1)/(2*j+1))*binomial(n,2*j)*binomial(n-1, 2*j) for j in range((n+1)//2))
%o [A136520(n) for n in range(1,41)] # _G. C. Greubel_, Jul 27 2023
%Y Cf. A001263, A027656.
%K nonn,easy
%O 1,3
%A _Gary W. Adamson_, Jan 02 2008
%E Terms a(11) onward added by _G. C. Greubel_, Jul 27 2023
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