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 A090238 #46 Oct 17 2022 07:22:26
%S A090238 1,0,1,0,2,1,0,6,4,1,0,24,16,6,1,0,120,72,30,8,1,0,720,372,152,48,10,
%T A090238 1,0,5040,2208,828,272,70,12,1,0,40320,14976,4968,1576,440,96,14,1,0,
%U A090238 362880,115200,33192,9696,2720,664,126,16,1,0,3628800,996480,247968,64704,17312,4380,952,160,18,1
%N A090238 Triangle T(n, k) read by rows. T(n, k) is the number of lists of k unlabeled permutations whose total length is n.
%C A090238 T(n,k) is the number of lists of k unlabeled permutations whose total length is n. Unlabeled means each permutation is on an initial segment of the positive integers. Example: with dashes separating permutations, T(3,2) = 4 counts 1-12, 1-21, 12-1, 21-1. - _David Callan_, Nov 29 2007
%C A090238 For n > 0, -Sum_{i=0..n} (-1)^i*T(n,i) is the number of indecomposable permutations A003319. - _Peter Luschny_, Mar 13 2009
%C A090238 Also the convolution triangle of the factorial numbers for n >= 1. - _Peter Luschny_, Oct 09 2022
%D A090238 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.
%F A090238 T(n, k) is given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
%F A090238 T(n, k) = T(n-1, k-1) + ((n+k-1)/k)*T(n-1, k); T(0, 0)=1, T(n, 0)=0 if n > 0, T(0, k)=0 if k > 0.
%F A090238 G.f. for the k-th column: (Sum_{i>=1} i!*t^i)^k = Sum_{n>=k} T(n, k)*t^n.
%F A090238 Sum_{k=0..n} T(n, k)*binomial(m, k) = A084938(m+n, m). - _Philippe Deléham_, Jan 31 2004
%F A090238 T(n, k) = Sum_{j>=0} A090753(j)*T(n-1, k+j-1). - _Philippe Deléham_, Feb 18 2004
%F A090238 From _Peter Bala_, May 27 2017: (Start)
%F A090238 Conjectural o.g.f.: 1/(1 + t - t*F(x)) = 1 + t*x + (2*t + t^2)*x^2 + (6*t + 4*t^2 + t^3)*x^3 + ..., where F(x) = Sum_{n >= 0} n!*x^n.
%F A090238 If true then a continued fraction representation of the o.g.f. is 1 - t + t/(1 - x/(1 - t*x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ... ))))))))). (End)
%e A090238 Triangle begins:
%e A090238 1;
%e A090238 0, 1;
%e A090238 0, 2, 1;
%e A090238 0, 6, 4, 1;
%e A090238 0, 24, 16, 6, 1;
%e A090238 0, 120, 72, 30, 8, 1;
%e A090238 0, 720, 372, 152, 48, 10, 1;
%e A090238 0, 5040, 2208, 828, 272, 70, 12, 1;
%e A090238 0, 40320, 14976, 4968, 1576, 440, 96, 14, 1;
%e A090238 0, 366880, 115200, 33192, 9696, 2720, 664, 126, 16, 1;
%e A090238 0, 3628800, 996480, 247968, 64704, 17312, 4380, 952, 160, 18, 1;
%e A090238 ...
%p A090238 T := proc(n,k) option remember; if n=0 and k=0 then return 1 fi;
%p A090238 if n>0 and k=0 or k>0 and n=0 then return 0 fi;
%p A090238 T(n-1,k-1)+(n+k-1)*T(n-1,k)/k end:
%p A090238 for n from 0 to 10 do seq(T(n,k),k=0..n) od; # _Peter Luschny_, Mar 03 2016
%p A090238 # Uses function PMatrix from A357368.
%p A090238 PMatrix(10, factorial); # _Peter Luschny_, Oct 09 2022
%t A090238 T[n_, k_] := T[n, k] = T[n-1, k-1] + ((n+k-1)/k)*T[n-1, k]; T[0, 0] = 1; T[_, 0] = T[0, _] = 0;
%t A090238 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 20 2018 *)
%Y A090238 Another version: A059369.
%Y A090238 Row sums: A051296, A003319 (n>0).
%Y A090238 Diagonals: A000007, A000142, A059371, A000012, A005843, A054000.
%Y A090238 Cf. A084938.
%K A090238 easy,nonn,tabl
%O A090238 0,5
%A A090238 _Philippe Deléham_, Jan 23 2004, Jun 14 2007
%E A090238 New name using a comment from _David Callan_ by _Peter Luschny_, Sep 01 2022