OFFSET
0,3
COMMENTS
Also the number of ballot sequences of length 2n such that the multiplicities of the largest and the smallest value differ by n.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..60
FORMULA
a(n) = A238707(2n,n).
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, k, l) `if`(n=0 or i<1 or `if`(l<>[], l[1], i)-1<k, 0,
`if`(l<>[] and l[1]-i=k, `if`(irem(n, i, 'j')=0, h([l[], i$j]),
0), add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i)))
end:
a:= n-> `if`(n=0, 1, g(2*n$2, n, [])):
seq(a(n), n=0..25);
MATHEMATICA
h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+
Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, l[[i]]}], {i, n}]];
g[n_, i_, k_, l_] := If[n == 0 || i<1 || If[l != {}, l[[1]], i]-1<k, 0,
If[l != {} && l[[1]] - i == k, j = Quotient[n, i];
If[Mod[n, i] == 0, h[Join[l, Table[i, {j}]]], 0],
Sum[g[n - i*j, i-1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]]];
a[n_] := If[n == 0, 1, g[2n, 2n, n, {}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 29 2021, after Maple code *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 25 2014
STATUS
approved