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Central terms of triangle A123610: a(n) = A123610(2*n,n).
6

%I #12 Sep 16 2024 23:58:43

%S 1,2,10,68,618,6352,71188,841332,10352618,131328068,1706742160,

%T 22619741212,304685855700,4160480013848,57476485976388,

%U 802048167035968,11290551106506218,160168176177137896,2287724464324213972

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

%C Related sequences: A123610(2n+1,n) = A000891(n); A123610(2n+2,n) = A123618(n); A123610(2n+2,n)/(n+1) = A123619(n).

%H G. C. Greubel, <a href="/A123617/b123617.txt">Table of n, a(n) for n = 0..830</a>

%t T[_, 0] = 1; T[n_, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]*Binomial[n/#, k/#]^2, 0] &];

%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* A123610 *)

%t Table[T[2*n, n], {n, 0, 50}] (* A123617 *)

%t Table[T[2*n + 2, n], {n, 0, 50}] (* A123618 *)

%t Table[T[2*n + 2,n]/(n+1), {n, 0, 50}] (* A123619 *)

%t (* _G. C. Greubel_, Oct 26 2017 *)

%o (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 Cf. A123610 (triangle); A000891, A123618.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 03 2006