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A013580 Triangle formed in same way as Pascal's triangle (A007318) except 1 is added to central element in even-numbered rows.

%I A013580 #20 Jul 02 2025 16:01:55
%S A013580 1,1,1,1,3,1,1,4,4,1,1,5,9,5,1,1,6,14,14,6,1,1,7,20,29,20,7,1,1,8,27,
%T A013580 49,49,27,8,1,1,9,35,76,99,76,35,9,1,1,10,44,111,175,175,111,44,10,1,
%U A013580 1,11,54,155,286,351,286,155,54,11,1,1,12,65,209,441,637,637,441,209,65
%N A013580 Triangle formed in same way as Pascal's triangle (A007318) except 1 is added to central element in even-numbered rows.
%C A013580 From _Gus Wiseman_, Apr 19 2023: (Start)
%C A013580 Appears to be the number of nonempty subsets of {1,...,n} with median k, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). For example, row n = 5 counts the following subsets:
%C A013580   {1}  {2}      {3}          {4}      {5}
%C A013580        {1,3}    {1,5}        {3,5}
%C A013580        {1,2,3}  {2,4}        {1,4,5}
%C A013580        {1,2,4}  {1,3,4}      {2,4,5}
%C A013580        {1,2,5}  {1,3,5}      {3,4,5}
%C A013580                 {2,3,4}
%C A013580                 {2,3,5}
%C A013580                 {1,2,4,5}
%C A013580                 {1,2,3,4,5}
%C A013580 Including half-steps gives A231147.
%C A013580 For mean instead of median we have A327481.
%C A013580 (End)
%H A013580 G. C. Greubel, <a href="/A013580/b013580.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A013580 G.f.: 1/(1-(1+y)*x)/(1-y*x^2). - _Vladeta Jovovic_, Oct 12 2003
%e A013580 Triangle begins:
%e A013580    1
%e A013580    1   1
%e A013580    1   3   1
%e A013580    1   4   4   1
%e A013580    1   5   9   5   1
%e A013580    1   6  14  14   6   1
%e A013580    1   7  20  29  20   7   1
%e A013580    1   8  27  49  49  27   8   1
%e A013580    1   9  35  76  99  76  35   9   1
%e A013580    1  10  44 111 175 175 111  44  10   1
%e A013580    1  11  54 155 286 351 286 155  54  11   1
%e A013580    1  12  65 209 441 637 637 441 209  65  12   1
%t A013580 CoefficientList[CoefficientList[Series[1/(1 - (1 + y)*x)/(1 - y*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* _G. C. Greubel_, Oct 10 2017 *)
%Y A013580 Row sums give A000975, A054106.
%Y A013580 Central diagonal T(2n+1,n+1) appears to be A006134.
%Y A013580 Central diagonal T(2n,n) appears to be A079309.
%Y A013580 For partitions instead of subsets we have A359901, row sums A325347.
%Y A013580 A000975 counts subsets with integer median.
%Y A013580 A007318 counts subsets by length, A359893 by twice median.
%Y A013580 Cf. A000984, A024718, A057552, A231147, A327475, A327481, A361654.
%K A013580 tabl,nonn,easy
%O A013580 0,5
%A A013580 Martin Hecko (bigusm(AT)interramp.com)
%E A013580 More terms from _James Sellers_