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A239119 Number of ballot sequences of length n with exactly 8 fixed points. 2
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 9, 29, 99, 357, 1351, 5343, 21992, 93912, 414848, 1891264, 8878972, 42849860, 212214460, 1077052284, 5594301872, 29704267536, 161055535088, 890880956848, 5022885935600, 28843306388880, 168562494708400, 1001888980299056 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

The fixed points are in the first 8 positions.

Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, ..., 8, but not 9.  An alternate definition uses the first row.

Conjecture: Generally, for fixed k is column k of A238802 asymptotic to sqrt(2)/(2*(k+1)*(k-1)!) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))), (holds for all k<=10). - Vaclav Kotesovec, Mar 11 2014

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>=10): (n-9)*(n^8 - 45*n^7 + 1302*n^6 - 34146*n^5 + 562989*n^4 - 4387005*n^3 + 7242668*n^2 + 80535276*n + 148594320)*a(n) = (n^9 - 54*n^8 + 1392*n^7 - 33705*n^6 + 734286*n^5 - 9696141*n^4 + 60317333*n^3 - 48716460*n^2 - 234532332*n - 4007057040)*a(n-1) + (n-10)*(n-8)*(n^8 - 37*n^7 + 1015*n^6 - 27223*n^5 + 410284*n^4 - 2451988*n^3 - 2863260*n^2 + 83948328*n + 232515360)*a(n-2). - Vaclav Kotesovec, Mar 11 2014

a(n) ~ sqrt(2)/90720 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1+7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 11 2014

EXAMPLE

a(8) = 1: [1,2,3,4,5,6,7,8].

a(9) = 1: [1,2,3,4,5,6,7,8,1].

a(10) = 3: [1,2,3,4,5,6,7,8,1,1], [1,2,3,4,5,6,7,8,1,2], [1,2,3,4,5,6,7,8,1,9].

a(11) = 9: [1,2,3,4,5,6,7,8,1,1,1], [1,2,3,4,5,6,7,8,1,1,2], [1,2,3,4,5,6,7,8,1,1,9], [1,2,3,4,5,6,7,8,1,2,1], [1,2,3,4,5,6,7,8,1,2,3], [1,2,3,4,5,6,7,8,1,2,9], [1,2,3,4,5,6,7,8,1,9,1], [1,2,3,4,5,6,7,8,1,9,2], [1,2,3,4,5,6,7,8,1,9,10].

MAPLE

b:= proc(n) option remember; `if`(n<3, [1, 1, 3][n+1],

      ((78*n^4 -18395*n^3 -71700*n^2 +536111*n -6824556)*b(n-1)

       +(203*n^5 +3335*n^4 +113400*n^3 +811949*n^2 -2733405*n

       +5461380)*b(n-2) +(n-3)*(125*n^4 +21309*n^3 +273479*n^2

       +556667*n +1829700)*b(n-3)) /

       (203*n^4+1789*n^3+80693*n^2+377071*n-3156156))

    end:

a:=n-> `if`(n<8, 0, b(n-8)):

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 == 8, 1, b[n - 9, {2, 1, 1, 1, 1, 1, 1, 1}]]; a[n_ /; n < 8] = 0; Table[ Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

CROSSREFS

Column k=8 of A238802.

Sequence in context: A239116 A239117 A239118 * A238803 A148940 A169781

Adjacent sequences:  A239116 A239117 A239118 * A239120 A239121 A239122

KEYWORD

nonn

AUTHOR

Joerg Arndt and Alois P. Heinz, Mar 10 2014

STATUS

approved

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Last modified June 19 12:57 EDT 2019. Contains 324222 sequences. (Running on oeis4.)