A238124 Number of ballot sequences of length n having exactly 1 largest part.

0, 1, 1, 3, 7, 20, 56, 182, 589, 2088, 7522, 28820, 113092, 464477, 1955760, 8541860, 38215077, 176316928, 832181774, 4033814912, 19973824386, 101257416701, 523648869394, 2765873334372, 14883594433742, 81646343582385, 455752361294076, 2589414185398032

Offset

0,4

Comments

Also number of standard Young tableaux with last row of length 1.

Column k=1 of A238123.

With different offset column k=2 of A238750.

Links

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..70

Example

The a(5)=20 ballot sequences of length 5 with 1 maximal element are (dots for zeros):
01:  [ . . . . 1 ]
02:  [ . . . 1 . ]
03:  [ . . . 1 2 ]
04:  [ . . 1 . . ]
05:  [ . . 1 . 2 ]
06:  [ . . 1 1 2 ]
07:  [ . . 1 2 . ]
08:  [ . . 1 2 1 ]
09:  [ . . 1 2 3 ]
10:  [ . 1 . . . ]
11:  [ . 1 . . 2 ]
12:  [ . 1 . 1 2 ]
13:  [ . 1 . 2 . ]
14:  [ . 1 . 2 1 ]
15:  [ . 1 . 2 3 ]
16:  [ . 1 2 . . ]
17:  [ . 1 2 . 1 ]
18:  [ . 1 2 . 3 ]
19:  [ . 1 2 3 . ]
20:  [ . 1 2 3 4 ]

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) `if`(n=0, 0, `if`(i=1, h([l[], 1$n]),
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=0..30);

Mathematica

b[n_, l_List] := b[n, l] = If[n < 1, x^l[[-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[0] = 0; a[n_] := Coefficient[b[n - 1, {1}], x, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 10 2015, after A238123 *)

Prog

(PARI) A238124(n)=A238123(n, 1) \\ M. F. Hasler, Jun 03 2018

Crossrefs

Cf. A238123, A238750.

Sequence in context: A245891 A058737 A274478 * A129429 A084204 A030238

Adjacent sequences: A238121 A238122 A238123 * A238125 A238126 A238127

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Status

approved