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A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%I A202064 #30 Jul 08 2025 07:41:59
%S A202064 1,2,0,3,1,0,4,4,0,0,5,10,1,0,0,6,20,6,0,0,0,7,35,21,1,0,0,0,8,56,56,
%T A202064 8,0,0,0,0,9,84,126,36,1,0,0,0,0,10,120,252,120,10,0,0,0,0,0,11,165,
%U A202064 462,330,55,1,0,0,0,0,0
%N A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A202064 Riordan array (x/(1-x)^2, x^2/(1-x)^2).
%C A202064 Mirror image of triangle in A119900.
%C A202064 A203322*A130595 as infinite lower triangular matrices. - _Philippe Deléham_, Jan 05 2011
%C A202064 From _Gus Wiseman_, Jul 07 2025: (Start)
%C A202064 Also the number of subsets of {1..n} containing n with k maximal runs (sequences of consecutive elements increasing by 1). For example, row n = 5 counts the following subsets:
%C A202064   {5}          {1,5}      {1,3,5}
%C A202064   {4,5}        {2,5}
%C A202064   {3,4,5}      {3,5}
%C A202064   {2,3,4,5}    {1,2,5}
%C A202064   {1,2,3,4,5}  {1,4,5}
%C A202064                {2,3,5}
%C A202064                {2,4,5}
%C A202064                {1,2,3,5}
%C A202064                {1,2,4,5}
%C A202064                {1,3,4,5}
%C A202064 For anti-runs instead of runs we have A053538.
%C A202064 Without requiring n see A210039, A202023, reverse A098158, A109446.
%C A202064 (End)
%F A202064 G.f.: 1/((1-x)^2-y*x^2).
%F A202064 Sum_{k, 0<=k<=n} T(n,k)*x^k = A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13 respectively.
%F A202064 T(n,k) = binomial(n+1,2k+1).
%F A202064 T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 15 2012
%e A202064 Triangle begins :
%e A202064 1
%e A202064 2, 0
%e A202064 3, 1, 0
%e A202064 4, 4, 0, 0
%e A202064 5, 10, 1, 0, 0
%e A202064 6, 20, 6, 0, 0, 0
%e A202064 7, 35, 21, 1, 0, 0, 0
%e A202064 8, 56, 56, 8, 0, 0, 0, 0
%t A202064 Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Length[Split[#,#2==#1+1&]]==k&]],{n,12},{k,n}] (* _Gus Wiseman_, Jul 07 2025 *)
%Y A202064 Cf. A007318, A005314 (antidiagonal sums), A119900, A084938, A130595, A203322.
%Y A202064 Column k = 1 is A000027.
%Y A202064 Row sums are A000079.
%Y A202064 Column k = 2 is A000292.
%Y A202064 Without zeros we have A034867.
%Y A202064 Last nonzero term in each row appears to be A124625.
%Y A202064 A034839 counts subsets by number of maximal runs, for anti-runs A384893.
%Y A202064 A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
%Y A202064 Cf. A000045, A000071, A001629, A010027, A053538, A208342, A210034, A245563, A268193, A384177, A384890.
%Y A202064 Cf. A098158, A109446, A202023, A210039.
%K A202064 nonn,tabl
%O A202064 0,2
%A A202064 _Philippe Deléham_, Dec 10 2011