cp's OEIS Frontend

A355010 Array read by ascending antidiagonals: T(n, k) is the number of n-core partitions with k corners.

%I A355010 #11 Jul 02 2022 14:51:28
%S A355010 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 A355010 76,13,1,36,280,721,771,396,106,15,1,45,456,1582,2331,1681,650,141,17,
%U A355010 1,55,705,3186,6244,6083,3235,995,181,19,1,66,1045,5985,15156,19348,13663,5685,1445,226,21,1
%N A355010 Array read by ascending antidiagonals: T(n, k) is the number of n-core partitions with k corners.
%C A355010 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 A355010 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 A355010 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 A355010 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 A355010 T(n, 2) = A006007(n-1).
%e A355010 The array begins:
%e A355010     1,  1,   1,   1,    1,    1,    1,    1, ...
%e A355010     3,  5,   7,   9,   11,   13,   15,   17, ...
%e A355010     6, 16,  31,  51,   76,  106,  141,  181, ...
%e A355010    10, 40, 105, 219,  396,  650,  995, 1445, ...
%e A355010    15, 85, 295, 771, 1681, 3235, 5685, 9325, ...
%e A355010    ...
%t A355010 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 A355010 Cf. A000012 (n = 2), A001263, A005408 (n = 3), A005891 (n = 4), A006007, A063490 (n = 5), A160747 (n = 6), A161680 (k = 1), A355011.
%K A355010 nonn,tabl
%O A355010 2,2
%A A355010 _Stefano Spezia_, Jun 15 2022