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A355010
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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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1
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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, 76, 13, 1, 36, 280, 721, 771, 396, 106, 15, 1, 45, 456, 1582, 2331, 1681, 650, 141, 17, 1, 55, 705, 3186, 6244, 6083, 3235, 995, 181, 19, 1, 66, 1045, 5985, 15156, 19348, 13663, 5685, 1445, 226, 21, 1
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OFFSET
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2,2
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COMMENTS
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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.).
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.
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LINKS
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FORMULA
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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.).
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EXAMPLE
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The array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
3, 5, 7, 9, 11, 13, 15, 17, ...
6, 16, 31, 51, 76, 106, 141, 181, ...
10, 40, 105, 219, 396, 650, 995, 1445, ...
15, 85, 295, 771, 1681, 3235, 5685, 9325, ...
...
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MATHEMATICA
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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}]]
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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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