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A239116
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Number of ballot sequences of length n with exactly 5 fixed points.
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2
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0, 0, 0, 0, 0, 1, 1, 3, 9, 29, 99, 356, 1343, 5279, 21584, 91324, 399456, 1799568, 8343404, 39702144, 193768604, 967992476, 4946617328, 25817913584, 137549830384, 747137750064, 4135349698416, 23301072909248, 133591802704944, 778722128953904, 4613070010373504
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OFFSET
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0,8
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COMMENTS
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The fixed points are in the first 5 positions.
Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, ..., 5, but not 6. An alternate definition uses the first row.
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LINKS
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FORMULA
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See Maple program.
Recurrence (for n>=7): (n-6)*(n^5 - 21*n^4 + 157*n^3 - 699*n^2 + 3298*n - 13680)*a(n) = (n^6 - 27*n^5 + 235*n^4 - 537*n^3 - 1964*n^2 - 2316*n + 54720)*a(n-1) + (n-7)*(n-5)*(n^5 - 16*n^4 + 83*n^3 - 344*n^2 + 2292*n - 10944)*a(n-2). - Vaclav Kotesovec, Mar 11 2014
a(n) ~ sqrt(2)/288 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 11 2014
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EXAMPLE
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a(5) = 1: [1,2,3,4,5].
a(6) = 1: [1,2,3,4,5,1].
a(7) = 3: [1,2,3,4,5,1,1], [1,2,3,4,5,1,2], [1,2,3,4,5,1,6].
a(8) = 9: [1,2,3,4,5,1,1,1], [1,2,3,4,5,1,1,2], [1,2,3,4,5,1,1,6], [1,2,3,4,5,1,2,1], [1,2,3,4,5,1,2,3], [1,2,3,4,5,1,2,6], [1,2,3,4,5,1,6,1], [1,2,3,4,5,1,6,2], [1,2,3,4,5,1,6,7].
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MAPLE
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a:= proc(n) option remember; `if`(n<6, [0$5, 1][n+1],
((952098*n^4 -28186656*n^3 +321186690*n^2 -1739275812*n
+3721544280)*a(n-1) +(n-7)*(451397*n^4 -9536389*n^3
+64448100*n^2 -229993164*n +534842280)*a(n-2)
-(n-7)*(n-8)*(500701*n^3 -9933473*n^2 +95681400*n
-319342500)*a(n-3))/
((n-5)*(451397*n^3-9487085*n^2+55580742*n-95239584)))
end:
seq(a(n), n=0..40);
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MATHEMATICA
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b[n_, l_List] := b[n, l] = If[n <= 0, 1, b[n - 1, Append[l, 1]] + Sum[If[i == 1 || l[[i - 1]] > l[[i]], b[n - 1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; a[n_] := If[n == 5, 1, b[n - 6, {2, 1, 1, 1, 1}]]; a[n_ /; n < 5] = 0; Table[ Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)
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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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