This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138771 #10 May 15 2017 08:10:24
%S A138771 1,1,1,2,3,1,6,11,5,2,24,50,26,14,6,120,274,154,94,54,24,720,1764,
%T A138771 1044,684,444,264,120,5040,13068,8028,5508,3828,2568,1560,720,40320,
%U A138771 109584,69264,49104,35664,25584,17520,10800,5040
%N A138771 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} whose 2nd cycle has k entries; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements (n>=1; 0<=k<=n-1). For example, 1432=(1)(24)(3) has 2 entries in the 2nd cycle; 3421=(1324) has 0 entries in the 2nd cycle.
%C A138771 T(n,0)=(n-1)!=A000142(n-1).
%C A138771 T(n,1)=A000254(n-1).
%C A138771 T(n,2)=A001705(n-2).
%C A138771 T(n,3)=2*A001711(n-4).
%C A138771 T(n,4)=6*A001716(n-5).
%C A138771 T(n,n-1)=(n-2)! (n>=2).
%C A138771 Sum(kT(n,k),k=0..n-1)=(n-1)!(n-1)(n+2)/4=A138772(n).
%F A138771 T(n,k)=(n-1)T(n-1,k)+(n-2)! (1<=k<=n-1). The row generating polynomials P[n](t) satisfy: P[n+1](t)=nP[n](t)+(n-1)!(t+t^2+...+t^n).
%e A138771 T(4,2)=5 because we have (1)(23)(4), (1)(24)(3), (13)(24), (12)(34) and (14)(23).
%e A138771 Triangle starts;
%e A138771 1;
%e A138771 1,1;
%e A138771 2,3,1;
%e A138771 6,11,5,2;
%e A138771 24,50,26,14,6;
%e A138771 120,274,154,94,54,24;
%p A138771 T:=proc (n,k) if k = 0 then factorial(n-1) elif n <= k then 0 else (n-1)*T(n-1, k)+factorial(n-2) end if end proc: for n to 9 do seq(T(n, k), k=0..n-1) end do;
%Y A138771 Cf. A000142, A000254, A001705, A001711, A001716, A138772.
%Y A138771 From _Johannes W. Meijer_, Oct 16 2009: (Start)
%Y A138771 A000142 equals for n=>1 the row sums.
%Y A138771 a(n) = A165680(n) * A165675(n-1).
%Y A138771 (End)
%K A138771 nonn,tabl
%O A138771 1,4
%A A138771 _Emeric Deutsch_, Apr 10 2008