A239116 Number of ballot sequences of length n with exactly 5 fixed points.

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

Offset

0,8

Comments

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.

Links

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..800

Wikipedia, Young tableau

Formula

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

Example

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].

Maple

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);

Mathematica

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 *)

Crossrefs

Column k=5 of A238802.

Sequence in context: A006134 A074526 A231291 * A239117 A239118 A239119

Adjacent sequences: A239113 A239114 A239115 * A239117 A239118 A239119

Keyword

nonn,easy

Author

Joerg Arndt and Alois P. Heinz, Mar 10 2014

Status

approved