cp's OEIS Frontend

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%I A202023 #26 Jul 08 2025 07:42:03
%S A202023 1,1,0,1,1,0,1,3,0,0,1,6,1,0,0,1,10,5,0,0,0,1,15,15,1,0,0,0,1,21,35,7,
%T A202023 0,0,0,0,1,28,70,28,1,0,0,0,0,1,36,126,84,9,0,0,0,0,0,1,45,210,210,45,
%U A202023 1,0,0,0,0,0
%N A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A202023 Riordan array (1/(1-x), x^2/(1-x)^2).
%C A202023 A skewed version of triangular array A085478.
%C A202023 Mirror image of triangle in A098158.
%C A202023 Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
%C A202023 Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
%C A202023 From _Gus Wiseman_, Jul 08 2025: (Start)
%C A202023 After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
%C A202023   {}  {1}        {1,3}    .  .  .
%C A202023       {2}        {1,4}
%C A202023       {3}        {2,4}
%C A202023       {4}        {1,2,4}
%C A202023       {1,2}      {1,3,4}
%C A202023       {2,3}
%C A202023       {3,4}
%C A202023       {1,2,3}
%C A202023       {2,3,4}
%C A202023       {1,2,3,4}
%C A202023 Requiring n-1 gives A202064.
%C A202023 For anti-runs instead of runs we have A384893.
%C A202023 (End)
%F A202023 T(n,k) = binomial(n,2k).
%F A202023 G.f.: (1-x)/((1-x)^2-y*x^2).
%F A202023 T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if n<k.
%F A202023 T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - _Philippe Deléham_, Nov 10 2013
%e A202023 Triangle begins :
%e A202023 1
%e A202023 1, 0
%e A202023 1, 1, 0
%e A202023 1, 3, 0, 0
%e A202023 1, 6, 1, 0, 0
%e A202023 1, 10, 5, 0, 0, 0
%e A202023 1, 15, 15, 1, 0, 0, 0
%e A202023 1, 21, 35, 7, 0, 0, 0, 0
%e A202023 1, 28, 70, 28, 1, 0, 0, 0, 0
%t A202023 Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* _Gus Wiseman_, Jul 08 2025 *)
%Y A202023 Column k = 1 is A000217.
%Y A202023 Column k = 2 is A000332.
%Y A202023 Row sums are A011782 (or A000079 shifted right).
%Y A202023 Removing all zeros gives A034839 (requiring n-1 A034867).
%Y A202023 Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
%Y A202023 Reversing rows gives A098158, without zeros A109446.
%Y A202023 Without the k = 0 column we get A210039.
%Y A202023 Row maxima appear to be A214282.
%Y A202023 A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
%Y A202023 A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
%Y A202023 Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.
%K A202023 nonn,tabl
%O A202023 0,8
%A A202023 _Philippe Deléham_, Dec 10 2011