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A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1.

%I A373889 #26 Jun 30 2024 22:01:06
%S A373889 2,1,3,1,2,4,1,2,4,6,1,2,4,7,9,1,2,3,6,11,14,1,2,3,6,10,17,22,1,2,3,5,
%T A373889 10,17,26,35,1,2,3,5,10,16,28,40,56,1,2,3,5,8,16,26,45,62,90,1,2,3,5,
%U A373889 8,16,26,43,71,97,145,1,2,3,5,8,13,26,42,71,111,153,234
%N A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1.
%H A373889 Paolo Xausa, <a href="/A373889/b373889.txt">Table of n, a(n) for n = 1..11325</a> (first 150 antidiagonals, flattened).
%H A373889 Hung Viet Chu and Zachary Louis Vasseur, <a href="https://arxiv.org/abs/2405.19352">Weighted Schreier-type Sets and the Fibonacci Sequence</a>, arXiv:2405.19352 [math.CO], 2024. See p. 2, Table 1 and Theorem 1.2.
%F A373889 T(k,n) = A000045(n-k+2) + 1, for k = 1 and n >= k;
%F A373889 T(k,n) = 2*(Sum_{i=0..k-2} binomial(n-k,i)*A000045(k-i)) + 2*binomial(n-k,k-1) + Sum_{j=1..n-k} binomial(j,n-j), for k >= 2 and n >= k;
%F A373889 T(k,n) = A000045(n+1) otherwise.
%F A373889 T(n,n) = 2*A000045(n).
%e A373889 The array begins:
%e A373889   k\n|  1  2  3  4   5   6   7   8   9   10  ...
%e A373889   ----------------------------------------------
%e A373889    1 |  2, 3, 4, 6,  9, 14, 22, 35, 56,  90, ... = A001611 (from n = 2).
%e A373889    2 |  1, 2, 4, 7, 11, 17, 26, 40, 62,  97, ...
%e A373889    3 |  1, 2, 4, 6, 10, 17, 28, 45, 71, 111, ...
%e A373889    4 |  1, 2, 3, 6, 10, 16, 26, 43, 71, 116, ...
%e A373889    5 |  1, 2, 3, 5, 10, 16, 26, 42, 68, 111, ...
%e A373889    6 |  1, 2, 3, 5,  8, 16, 26, 42, 68, 110, ...
%e A373889    7 |  1, 2, 3, 5,  8, 13, 26, 42, 68, 110, ...
%e A373889    8 |  1, 2, 3, 5,  8, 13, 21, 42, 68, 110, ...
%e A373889    9 |  1, 2, 3, 5,  8, 13, 21, 34, 68, 110, ...
%e A373889   10 |  1, 2, 3, 5,  8, 13, 21, 34, 55, 110, ...
%e A373889   ...
%t A373889 A373889[k_, n_] := Which[n < k, Fibonacci[n+1], k == 1, Fibonacci[n-k+2] + 1, True, 2*Sum[Binomial[n-k, i]*Fibonacci[k-i], {i, 0, k-2}] + 2*Binomial[n-k, k-1] + Sum[Binomial[j, n-j], {j, n-k}]];
%t A373889 Table[A373889[k-n+1, n], {k, 15}, {n, k}]
%Y A373889 Cf. A000045, A001611, A373345, A373556, A373557.
%K A373889 nonn,tabl
%O A373889 1,1
%A A373889 _Paolo Xausa_, Jun 21 2024