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A359067 a(2*n) = Sum_{k=0..n-1} binomial(2*n,k) binomial(2*n-1-k, n-1-k). a(2*n+1) = (Sum_{k=0..n} binomial(2*n+1,k) binomial(2*n-k, n-k)) - binomial(2*n-1, n).

%I A359067 #24 Jan 09 2023 16:41:55
%S A359067 0,1,4,7,28,49,199,351,1436,2561,10499,18943,77617,141569,579149,
%T A359067 1066495,4354780,8085505,32954635,61616127,250713893,471556097,
%U A359067 1915928117,3621830655,14696701553,27902803969,113099318869,215530668031,872780984131,1668644405249,6751457741849
%N A359067 a(2*n) = Sum_{k=0..n-1} binomial(2*n,k) binomial(2*n-1-k, n-1-k). a(2*n+1) = (Sum_{k=0..n} binomial(2*n+1,k) binomial(2*n-k, n-k)) - binomial(2*n-1, n).
%C A359067 For n >= 3, the number of admissible pinnacle sets in the group S_n^D of even-signed permutations.
%C A359067 The even-indexed terms match the even-indexed terms of A359066. The odd-indexed terms differ from the odd-indexed terms of A359066 by binomial(2*n-1, n).
%H A359067 Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, <a href="https://arxiv.org/abs/2301.02628">Pinnacle sets of signed permutations</a>, arXiv:2301.02628 [math.CO] (2023).
%F A359067 a(2*n) = Sum_{k=0..n-1} binomial(2*n,k) binomial(2*n-1-k, n-1-k).
%F A359067 a(2*n+1) = (Sum_{k=0..n} binomial(2*n+1,k) binomial(2*n-k, n-k)) - binomial(2*n-1, n).
%F A359067 a(n) = A240721((n-2)/2) if n-1 is odd and otherwise A178792((n-1)/2) - binomial(2*n - 1, n). - _Peter Luschny_, Jan 03 2023
%e A359067 For n = 3, the a(3) = 4 admissible pinnacle sets in S_3^D are {}, {1}, {2}, {3}.
%p A359067 a := n -> if irem(n - 1, 2) = 1 then binomial(n, n/2 - 1)*hypergeom([n/2 + 1, -n/2 + 1], [n/2 + 2], -1) else binomial(n + 1, n/2 + 1/2)*hypergeom([n/2 + 1/2, -n/2 + 1/2], [n/2 + 3/2], -1)/2 - binomial(n - 2, n/2 - 1/2) fi:
%p A359067 seq(simplify(a(n)), n = 3..31); # _Peter Luschny_, Jan 03 2023
%Y A359067 Cf. A289871, A359066, A359068, A178792, A240721.
%K A359067 easy,nonn
%O A359067 1,3
%A A359067 _Bridget Tenner_, Dec 15 2022