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A359066 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).

%I A359066 #36 Jan 09 2023 21:30:51
%S A359066 1,1,5,7,31,49,209,351,1471,2561,10625,18943,78079,141569,580865,
%T A359066 1066495,4361215,8085505,32978945,61616127,250806271,471556097,
%U A359066 1916280833,3621830655,14698053631,27902803969,113104519169,215530668031,872801042431,1668644405249,6751535300609
%N A359066 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).
%C A359066 For n >= 3, this is the number of admissible pinnacle sets in the group S_n^B of signed permutations.
%C A359066 The even-indexed terms appear in A240721 and the odd-indexed terms appear in A178792.
%H A359066 Andrew Howroyd, <a href="/A359066/b359066.txt">Table of n, a(n) for n = 1..1000</a>
%H A359066 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 A359066 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).
%F A359066 a(n) = binomial(n-1, floor((n-1)/2))*hypergeom([-n, ceil((1 -n)/2)], [1 - n], -1). - _Peter Luschny_, Jan 03 2023
%e A359066 For n = 3, the a(3) = 5 admissible pinnacle sets in S_3^B are {}, {-1}, {1}, {2}, {3}.
%p A359066 a := n -> add(binomial(n, k)*binomial(n-1-k, iquo(n-1, 2) - k), k = 0..iquo(n-1,2)):
%p A359066 # Alternative:
%p A359066 a := n -> binomial(n-1, floor((n-1)/2))*hypergeom([-n, ceil((1-n)/2)], [1-n], -1);
%p A359066 seq(simplify(a(n)), n=3..31); # _Peter Luschny_, Jan 03 2023
%t A359066 Array[Sum[Binomial[#, k]*Binomial[# - 1 - k, Floor[(# - 1)/2] - k], {k, 0, Floor[(# - 1)/2]}] &, 31] (* _Michael De Vlieger_, Jan 03 2023 *)
%o A359066 (PARI) a(n) = sum(k=0, (n-1)\2, binomial(n,k)*binomial(n-1-k, (n-1)\2 - k)) \\ _Andrew Howroyd_, Jan 02 2023
%Y A359066 Cf. A289871, A359067, A359068, A240721, A178792.
%K A359066 easy,nonn
%O A359066 1,3
%A A359066 _Bridget Tenner_, Dec 15 2022