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A245001 Number of standard Young tableaux with n cells and 3 as last value in the first row. 2
2, 3, 5, 10, 19, 41, 86, 197, 449, 1087, 2650, 6722, 17227, 45267, 120069, 323442, 877777, 2405399, 6628760, 18384040, 51204735, 143252991, 402115301, 1132464571, 3197928097, 9053803101, 25689876776, 73047889402, 208100836969, 593897902349, 1697686011406 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

Also the number of ballot sequences of length n where 3 is the position of the last occurrence of the minimal value.

LINKS

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 3..1000

Wikipedia, Young tableau

FORMULA

Recurrence: see Maple program.

a(n) ~ 3^(n-3/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 11 2014

EXAMPLE

a(4) = 3:

[1 3]  [1 3]  [1 2 3]

[2]    [2 4]  [4]

[4]

MAPLE

a:= proc(n) option remember; `if`(n<5, [0$3, 2, 3][n+1],

     ((n-2)*(30*n^4-505*n^3+3108*n^2-8147*n+7338)*a(n-1)

     +(18703*n^3-76648*n^2+154520*n-122616-2240*n^4+105*n^5)*a(n-2)

     -2*(n-5)*(60*n^4-965*n^3+5766*n^2-15082*n+14364)*a(n-3)

     -12*(n-5)*(n-6)*(15*n^3-185*n^2+744*n-994)*a(n-4)) /

     ((n-1)*(n-2)*(15*n^3-230*n^2+1159*n-1938)))

    end:

seq(a(n), n=3..40);

MATHEMATICA

b[n_, l_List] := b[n, l] = If[n == 0, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + Function[{p}, p+(x^(1 + Total[l])-1)*Coefficient[p, x, 0]][b[n-1, Append[l, 1]]]]; a[n_] := Coefficient[ b[n, {}], x, 3]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 3, 40}] (* Jean-Fran├žois Alcover, Feb 06 2015, after Maple code in A238794 *)

CROSSREFS

Column k=3 of A238794.

Sequence in context: A293353 A293328 A064236 * A007569 A054317 A065840

Adjacent sequences:  A244998 A244999 A245000 * A245002 A245003 A245004

KEYWORD

nonn

AUTHOR

Joerg Arndt and Alois P. Heinz, Jul 09 2014

STATUS

approved

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Last modified November 17 04:01 EST 2018. Contains 317275 sequences. (Running on oeis4.)