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A123619 a(n) = A123610(2*n+2,n)/(n+1) = A123618(n)/(n+1).

%I A123619 #6 Oct 26 2017 16:45:13
%S A123619 1,2,13,98,884,8712,92033,1022450,11819620,141052808,1727897780,
%T A123619 21634496072,275950213712,3576314656800,46995009879033,
%U A123619 625082413914450,8403885788094500,114069363868845000,1561609591376307572
%N A123619 a(n) = A123610(2*n+2,n)/(n+1) = A123618(n)/(n+1).
%C A123619 Related sequences: A123610(2n,n) = A123617(n); A123610(2n+1,n) = A000891(n); A123610(2n+2,n) = A123618(n).
%H A123619 G. C. Greubel, <a href="/A123619/b123619.txt">Table of n, a(n) for n = 0..830</a>
%t A123619 T[_, 0] = 1; T[n_, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]*Binomial[n/#, k/#]^2, 0] &];
%t A123619 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* A123610 *)
%t A123619 Table[T[2*n, n], {n, 0, 50}] (* A123617 *)
%t A123619 Table[T[2*n + 2, n], {n, 0, 50}] (* A123618 *)
%t A123619 Table[T[2*n + 2,n]/(n+1), {n, 0, 50}] (* A123619 *)
%t A123619 (* _G. C. Greubel_, Oct 26 2017 *)
%o A123619 (PARI) {a(n)=if(n==0,1,(1/(2*(n+1)^2))*sumdiv(2*n+2,d,if(gcd(n,d)==d, eulerphi(d)*binomial((2*n+2)/d,n/d)^2,0)))}
%Y A123619 Cf. A123610 (triangle); A123617, A000891, A123618.
%K A123619 nonn
%O A123619 0,2
%A A123619 _Paul D. Hanna_, Oct 03 2006