cp's OEIS Frontend

A355011 Array read by ascending antidiagonals: T(n, k) is the number of self-conjugate n-core partitions with k corners.

%I A355011 #13 Jul 02 2022 14:51:47
%S A355011 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 A355011 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 A355011 76,124,55,69,13,14,1,1,6,36,65,175,124,216,69,93,14,16,1,1
%N A355011 Array read by ascending antidiagonals: T(n, k) is the number of self-conjugate n-core partitions with k corners.
%C A355011 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 A355011 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 A355011 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 A355011 T(4, n) = T(5, n) = A001651(n+1).
%e A355011 The array begins:
%e A355011   1,  1,  1,   1,   1,   1,    1,    1, ...
%e A355011   1,  1,  1,   1,   1,   1,    1,    1, ...
%e A355011   2,  4,  5,   7,   8,  10,   11,   13, ...
%e A355011   2,  4,  5,   7,   8,  10,   11,   13, ...
%e A355011   3,  9, 15,  27,  37,  55,   69,   93, ...
%e A355011   3,  9, 15,  27,  37,  55,   69,   93, ...
%e A355011   4, 16, 34,  76, 124, 216,  309,  471, ...
%e A355011   4, 16, 34,  76, 124, 216,  309,  471, ...
%e A355011   5, 25, 65, 175, 335, 675, 1095, 1875, ...
%e A355011   5, 25, 65, 175, 335, 675, 1095, 1875, ...
%e A355011   ...
%t A355011 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 A355011 Cf. A000012 (n = 2,3), A001651, A004526 (k = 1), A008794 (k = 2), A247643 (n = 6,7), A355010.
%K A355011 nonn,tabl
%O A355011 2,4
%A A355011 _Stefano Spezia_, Jun 15 2022