OFFSET
5,1
COMMENTS
Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 3.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 5..150
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) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
l[1]-i=3, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
end:
a:= n-> g(n$2, []):
seq(a(n), n=5..35);
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, [[i]]}], {i, n}]];
g[n_, i_, l_] := Module[{j}, If[n == 0 || i < 1, 0, If[l != {} && l[[1]] - i == 3, If[j = Quotient[n, i]; Mod[n, i] == 0, h[Join[l, Table[i, {j}]]], 0], Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]]]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 5, 35}] (* Jean-François Alcover, Aug 28 2021, after Maple code *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 25 2014
STATUS
approved