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A373889
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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.
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1
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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, 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, 8, 16, 26, 43, 71, 97, 145, 1, 2, 3, 5, 8, 13, 26, 42, 71, 111, 153, 234
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OFFSET
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1,1
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LINKS
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FORMULA
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T(k,n) = A000045(n-k+2) + 1, for k = 1 and n >= k;
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;
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EXAMPLE
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The array begins:
k\n| 1 2 3 4 5 6 7 8 9 10 ...
----------------------------------------------
1 | 2, 3, 4, 6, 9, 14, 22, 35, 56, 90, ... = A001611 (from n = 2).
2 | 1, 2, 4, 7, 11, 17, 26, 40, 62, 97, ...
3 | 1, 2, 4, 6, 10, 17, 28, 45, 71, 111, ...
4 | 1, 2, 3, 6, 10, 16, 26, 43, 71, 116, ...
5 | 1, 2, 3, 5, 10, 16, 26, 42, 68, 111, ...
6 | 1, 2, 3, 5, 8, 16, 26, 42, 68, 110, ...
7 | 1, 2, 3, 5, 8, 13, 26, 42, 68, 110, ...
8 | 1, 2, 3, 5, 8, 13, 21, 42, 68, 110, ...
9 | 1, 2, 3, 5, 8, 13, 21, 34, 68, 110, ...
10 | 1, 2, 3, 5, 8, 13, 21, 34, 55, 110, ...
...
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MATHEMATICA
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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}]];
Table[A373889[k-n+1, n], {k, 15}, {n, k}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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