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

Author

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Keywords

• nonn

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.

Examples

			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 ]
		

Links

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

Crossrefs

Cf. A238123, A238750.

Programs

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 *)

PARI

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