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

%I A238124 #27 Jun 04 2018 10:00:30
%S A238124 0,1,1,3,7,20,56,182,589,2088,7522,28820,113092,464477,1955760,
%T A238124 8541860,38215077,176316928,832181774,4033814912,19973824386,
%U A238124 101257416701,523648869394,2765873334372,14883594433742,81646343582385,455752361294076,2589414185398032
%N A238124 Number of ballot sequences of length n having exactly 1 largest part.
%C A238124 Also number of standard Young tableaux with last row of length 1.
%C A238124 Column k=1 of A238123.
%C A238124 With different offset column k=2 of A238750.
%H A238124 Joerg Arndt and Alois P. Heinz, <a href="/A238124/b238124.txt">Table of n, a(n) for n = 0..70</a>
%e A238124 The a(5)=20 ballot sequences of length 5 with 1 maximal element are (dots for zeros):
%e A238124 01:  [ . . . . 1 ]
%e A238124 02:  [ . . . 1 . ]
%e A238124 03:  [ . . . 1 2 ]
%e A238124 04:  [ . . 1 . . ]
%e A238124 05:  [ . . 1 . 2 ]
%e A238124 06:  [ . . 1 1 2 ]
%e A238124 07:  [ . . 1 2 . ]
%e A238124 08:  [ . . 1 2 1 ]
%e A238124 09:  [ . . 1 2 3 ]
%e A238124 10:  [ . 1 . . . ]
%e A238124 11:  [ . 1 . . 2 ]
%e A238124 12:  [ . 1 . 1 2 ]
%e A238124 13:  [ . 1 . 2 . ]
%e A238124 14:  [ . 1 . 2 1 ]
%e A238124 15:  [ . 1 . 2 3 ]
%e A238124 16:  [ . 1 2 . . ]
%e A238124 17:  [ . 1 2 . 1 ]
%e A238124 18:  [ . 1 2 . 3 ]
%e A238124 19:  [ . 1 2 3 . ]
%e A238124 20:  [ . 1 2 3 4 ]
%p A238124 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A238124        add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A238124     end:
%p A238124 g:= proc(n, i, l) `if`(n=0, 0, `if`(i=1, h([l[], 1$n]),
%p A238124        add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
%p A238124     end:
%p A238124 a:= n-> g(n, n, []):
%p A238124 seq(a(n), n=0..30);
%t A238124 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 *)
%o A238124 (PARI) A238124(n)=A238123(n,1) \\ _M. F. Hasler_, Jun 03 2018
%Y A238124 Cf.  A238123, A238750.
%K A238124 nonn
%O A238124 0,4
%A A238124 _Joerg Arndt_ and _Alois P. Heinz_, Feb 21 2014