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Triangle formed in same way as Pascal's triangle (A007318) except 1 is added to central element in even-numbered rows.
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%I #19 Apr 20 2023 14:56:14

%S 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 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 1,11,54,155,286,351,286,155,54,11,1,1,12,65,209,441,637,637,441,209,65

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

%C From _Gus Wiseman_, Apr 19 2023: (Start)

%C 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 {1} {2} {3} {4} {5}

%C {1,3} {1,5} {3,5}

%C {1,2,3} {2,4} {1,4,5}

%C {1,2,4} {1,3,4} {2,4,5}

%C {1,2,5} {1,3,5} {3,4,5}

%C {2,3,4}

%C {2,3,5}

%C {1,2,4,5}

%C {1,2,3,4,5}

%C Including half-steps gives A231147.

%C For mean instead of median we have A327481.

%C (End)

%H G. C. Greubel, <a href="/A013580/b013580.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F G.f.: 1/(1-(1+y)*x)/(1-y*x^2). - _Vladeta Jovovic_, Oct 12 2003

%e Triangle begins:

%e 1

%e 1 1

%e 1 3 1

%e 1 4 4 1

%e 1 5 9 5 1

%e 1 6 14 14 6 1

%e 1 7 20 29 20 7 1

%e 1 8 27 49 49 27 8 1

%e 1 9 35 76 99 76 35 9 1

%e 1 10 44 111 175 175 111 44 10 1

%e 1 11 54 155 286 351 286 155 54 11 1

%e 1 12 65 209 441 637 637 441 209 65 12 1

%t 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 Row sums give A000975, A054106.

%Y Central diagonal T(2n+1,n+1) appears to be A006134.

%Y Central diagonal T(2n,n) appears to be A079309.

%Y For partitions instead of subsets we have A359901, row sums A325347.

%Y A000975 counts subsets with integer median.

%Y A007318 counts subsets by length, A359893 by twice median.

%Y Cf. A000984, A024718, A057552, A231147, A327475, A327481, A361654.

%K tabl,nonn,easy

%O 0,5

%A Martin Hecko (bigusm(AT)interramp.com)

%E More terms from _James A. Sellers_