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%I #13 Jul 02 2022 14:51:47
%S 1,1,1,2,1,1,2,4,1,1,3,4,5,1,1,3,9,5,7,1,1,4,9,15,7,8,1,1,4,16,15,27,
%T 8,10,1,1,5,16,34,27,37,10,11,1,1,5,25,34,76,37,55,11,13,1,1,6,25,65,
%U 76,124,55,69,13,14,1,1,6,36,65,175,124,216,69,93,14,16,1,1
%N Array read by ascending antidiagonals: T(n, k) is the number of self-conjugate n-core partitions with k corners.
%C T(n, k) is also equal to the number of cornerless symmetric Motzkin paths of length 2*k + n - 1 with n - 1 flat steps (see Theorem 3.7 and Proposition 3.8 at pp. 16 - 17 in Cho et al.).
%H Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, <a href="https://arxiv.org/abs/2205.15554">Combinatorics on bounded free Motzkin paths and its applications</a>, arXiv:2205.15554 [math.CO], 2022.
%F T(n, k) = Sum_{i=1..min(k,floor(n/2))} binomial(floor((k-1)/2), floor((i-1)/2))*binomial(floor(k/2), floor(i/2))*binomial(floor(n/2)+k-i, k). (See proposition 3.8 in Cho et al.).
%F T(4, n) = T(5, n) = A001651(n+1).
%e The array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 2, 4, 5, 7, 8, 10, 11, 13, ...
%e 2, 4, 5, 7, 8, 10, 11, 13, ...
%e 3, 9, 15, 27, 37, 55, 69, 93, ...
%e 3, 9, 15, 27, 37, 55, 69, 93, ...
%e 4, 16, 34, 76, 124, 216, 309, 471, ...
%e 4, 16, 34, 76, 124, 216, 309, 471, ...
%e 5, 25, 65, 175, 335, 675, 1095, 1875, ...
%e 5, 25, 65, 175, 335, 675, 1095, 1875, ...
%e ...
%t T[n_,k_]:=Sum[Binomial[Floor[(k-1)/2],Floor[(i-1)/2]]Binomial[Floor[k/2],Floor[i/2]]Binomial[Floor[n/2]+k-i,k],{i,Min[k,Floor[n/2]]}]; Flatten[Table[T[n-k+1,k],{n,2,13},{k,1,n-1}]]
%Y Cf. A000012 (n = 2,3), A001651, A004526 (k = 1), A008794 (k = 2), A247643 (n = 6,7), A355010.
%K nonn,tabl
%O 2,4
%A _Stefano Spezia_, Jun 15 2022