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A210034 Triangle of coefficients of polynomials v(n,x) jointly generated with A210033; see the Formula section.

%I A210034 #13 Aug 07 2025 04:37:57
%S A210034 1,2,1,4,2,1,7,5,2,1,12,10,6,2,1,20,20,13,7,2,1,33,38,29,16,8,2,1,54,
%T A210034 71,60,39,19,9,2,1,88,130,122,86,50,22,10,2,1,143,235,241,187,116,62,
%U A210034 25,11,2,1,232,420,468,392,267,150,75,28,12,2,1,376,744,894,806
%N A210034 Triangle of coefficients of polynomials v(n,x) jointly generated with A210033; see the Formula section.
%C A210034 For a discussion and guide to related arrays, see A208510.
%C A210034 From _Gus Wiseman_, Jun 29 2025: (Start)
%C A210034 This appears to be the number of subsets of {1..n} with k>0 maximal anti-runs (sequences of consecutive elements increasing by more than 1). For example, the subset {1,2,4,5} has maximal anti-runs ((1),(2,4),(5)) so is counted under T(5,3). Row n = 5 counts the following:
%C A210034   {1}      {1,2}    {1,2,3}    {1,2,3,4}  {1,2,3,4,5}
%C A210034   {2}      {2,3}    {2,3,4}    {2,3,4,5}
%C A210034   {3}      {3,4}    {3,4,5}
%C A210034   {4}      {4,5}    {1,2,3,5}
%C A210034   {5}      {1,2,4}  {1,2,4,5}
%C A210034   {1,3}    {1,2,5}  {1,3,4,5}
%C A210034   {1,4}    {1,3,4}
%C A210034   {1,5}    {1,4,5}
%C A210034   {2,4}    {2,3,5}
%C A210034   {2,5}    {2,4,5}
%C A210034   {3,5}
%C A210034   {1,3,5}
%C A210034 For runs instead of anti-runs we have A034839, with n A202064. For reversed partitions instead of subsets we have A268193. (End)
%F A210034 u(n,x)=u(n-1,x)+v(n-1,x)+1,
%F A210034 v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
%F A210034 where u(1,x)=1, v(1,x)=1.
%e A210034 First five rows:
%e A210034   1
%e A210034   2    1
%e A210034   4    2    1
%e A210034   7    5    2   1
%e A210034   12   10   6   2   1
%e A210034 First three polynomials v(n,x): 1, 2 + x, 4 + 2*x + x^2.
%t A210034 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210034 u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A210034 v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210034 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210034 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210034 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210034 TableForm[cu]
%t A210034 Flatten[%]    (* A210033 *)
%t A210034 Table[Expand[v[n, x]], {n, 1, z}]
%t A210034 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210034 TableForm[cv]
%t A210034 Flatten[%]    (* A210034 *)
%Y A210034 Cf. A210033, A208510.
%Y A210034 Column k = 1 is A000071.
%Y A210034 Row sums are A000225.
%Y A210034 Column k = 2 is A001629.
%Y A210034 Column k = 3 is A055243.
%Y A210034 The version including k = 0 is A384893.
%Y A210034 A034839 counts subsets by number of maximal runs, see also A202023, A202064.
%Y A210034 A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
%Y A210034 A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
%Y A210034 Cf. A053538, A116674, A119900, A268193, A384177, A384879, A384889, A384905.
%K A210034 nonn,tabl
%O A210034 1,2
%A A210034 _Clark Kimberling_, Mar 16 2012