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A123617 Central terms of triangle A123610: a(n) = A123610(2*n,n).

%I A123617 #12 Sep 16 2024 23:58:43
%S A123617 1,2,10,68,618,6352,71188,841332,10352618,131328068,1706742160,
%T A123617 22619741212,304685855700,4160480013848,57476485976388,
%U A123617 802048167035968,11290551106506218,160168176177137896,2287724464324213972
%N A123617 Central terms of triangle A123610: a(n) = A123610(2*n,n).
%C A123617 Related sequences: A123610(2n+1,n) = A000891(n); A123610(2n+2,n) = A123618(n); A123610(2n+2,n)/(n+1) = A123619(n).
%H A123617 G. C. Greubel, <a href="/A123617/b123617.txt">Table of n, a(n) for n = 0..830</a>
%t A123617 T[_, 0] = 1; T[n_, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]*Binomial[n/#, k/#]^2, 0] &];
%t A123617 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* A123610 *)
%t A123617 Table[T[2*n, n], {n, 0, 50}] (* A123617 *)
%t A123617 Table[T[2*n + 2, n], {n, 0, 50}] (* A123618 *)
%t A123617 Table[T[2*n + 2,n]/(n+1), {n, 0, 50}] (* A123619 *)
%t A123617 (* _G. C. Greubel_, Oct 26 2017 *)
%o A123617 (PARI) {a(n)=if(n==0,1,(1/2/n)*sumdiv(2*n,d,if(gcd(n,d)==d, eulerphi(d)*binomial(2*n/d,n/d)^2,0)))}
%Y A123617 Cf. A123610 (triangle); A000891, A123618.
%K A123617 nonn
%O A123617 0,2
%A A123617 _Paul D. Hanna_, Oct 03 2006