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A308679
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Kuba-Panholzer Table 2 pattern 123, 321 for Stirling permutation k = 2.
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0
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5, 33, 180, 919, 4560, 22332, 108733, 528298, 2566516, 12480403, 60784064, 296593256, 1450124169, 7104618375, 34878823088, 171572357252, 845605268800, 4175311417840, 20652607880698, 102326794307846, 507804406403540, 2523838310290891, 12561785900116608, 62608677333571728
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OFFSET
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3,1
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LINKS
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FORMULA
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For k = 2, Sum_{j, 0..k} (C(k, j) * (-1)^j * ((k - 1) * Sum_{m = 0..(n - j)} (C(n - j - 1 + (k - 1) m, n - j - m) (2 C(n - j + 2, m)/(n - j + 2) - C(n - j + 1, m)/(n - j + 1))) + Sum_{m = 0..n + 1 - j} C(n - j + (k - 1) m, n - j + 1 - m) * (2 C(n + 1 - j + 2, m)/(n + 1 - j + 2) - C(n + 1 - j + 1, m)/(n + 1 - j + 1)))) + (2 (2 - k))/(n + 1) Sum_{{m = 0..(n - 1)} (C(n + 1, m) * C(n - 2 + (k + 1) m, n - 1 - m)) - (4/(n + 2)) * Sum_{m = 0..n} (C(n + 2, m) * C(n - 1 + (k - 1) m, n - m)) + Sum_{m = 0..n} (C(n, m) * C(n + (k - 1) m - 1, n - m)/(n + 1 - m)) (See Kuba-Panholzer paper).
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MATHEMATICA
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With[{k = 2}, Table[Sum[Binomial[k, j] (-1)^j*((k - 1) Sum[Binomial[n - j - 1 + (k - 1) m, n - j - m] (2 Binomial[#, m]/# &[n - j + 2] - Binomial[#, m]/# &[n - j + 1]), {m, 0, n - j}] + Sum[Binomial[n - j + (k - 1) m, n - j + 1 - m] (2 Binomial[#, m]/# &[n + 1 - j + 2] - Binomial[#, m]/# &[n + 1 - j + 1]), {m, 0, n + 1 - j}]), {j, 0, k}] + (2 (2 - k))/(n + 1) Sum[Binomial[n + 1, m] Binomial[n - 2 + (k + 1) m, n - 1 - m], {m, 0, n - 1}] - (4/(n + 2)) Sum[Binomial[n + 2, m] Binomial[n - 1 + (k - 1) m, n - m], {m, 0, n}] + Sum[Binomial[n, m] Binomial[n + (k - 1) m - 1, n - m]/(n + 1 - m), {m, 0, n}], {n, 0, 25}]]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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