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Array read by ascending antidiagonals: T(n, k) is the number of n-core partitions with k corners.
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%I #11 Jul 02 2022 14:51:28

%S 1,3,1,6,5,1,10,16,7,1,15,40,31,9,1,21,85,105,51,11,1,28,161,295,219,

%T 76,13,1,36,280,721,771,396,106,15,1,45,456,1582,2331,1681,650,141,17,

%U 1,55,705,3186,6244,6083,3235,995,181,19,1,66,1045,5985,15156,19348,13663,5685,1445,226,21,1

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

%C T(n, k) is also equal to the number of cornerless Motzkin paths of length 2*k + n - 1 with n - 1 flat steps (see Theorem 3.3 and Proposition 3.4 at pp. 13 - 14 in Cho et al.).

%C In proposition 3.4 in Cho et al., the Narayana number is defined as N(k, i) = binomial(k, i)*binomial(k, i-1)/k, unlike A001263.

%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))} N(k, i)*binomial(n+2*(k-i), 2*k), where N(k, i) = binomial(k, i)*binomial(k, i-1)/k. (See proposition 3.4 in Cho et al.).

%F T(n, 2) = A006007(n-1).

%e The array begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 3, 5, 7, 9, 11, 13, 15, 17, ...

%e 6, 16, 31, 51, 76, 106, 141, 181, ...

%e 10, 40, 105, 219, 396, 650, 995, 1445, ...

%e 15, 85, 295, 771, 1681, 3235, 5685, 9325, ...

%e ...

%t T[n_,k_]:=Sum[Binomial[k,i]Binomial[k,i-1]Binomial[n+2(k-i),2k]/k,{i,Min[k,Floor[n/2]]}]; Flatten[Table[T[n-k+1,k],{n,2,12},{k,1,n-1}]]

%Y Cf. A000012 (n = 2), A001263, A005408 (n = 3), A005891 (n = 4), A006007, A063490 (n = 5), A160747 (n = 6), A161680 (k = 1), A355011.

%K nonn,tabl

%O 2,2

%A _Stefano Spezia_, Jun 15 2022