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
Keywords
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
Programs
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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
Comments