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%I #16 Aug 25 2021 13:04:45
%S 2,3,14,14,69,97,251,671,1847,2111,12869,33461,58343,189045,841125,
%T 2207347,6651215,12781755,73096191,308508927,904926489,1727792245,
%U 7638794959,44017642189,177969495449,522668483639,1662245807549,4496811662189,32142974215379
%N Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 1.
%C Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 1.
%H Alois P. Heinz, <a href="/A244295/b244295.txt">Table of n, a(n) for n = 3..400</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e a(4) = 3:
%e [1 2] [1 3] [1 4]
%e [3] [2] [2]
%e [4] [4] [3]
%p h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p end:
%p g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
%p l[1]-i=1, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
%p add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
%p end:
%p a:= n-> g(n, n, []):
%p seq(a(n), n=3..35);
%t 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}]];
%t g[n_, i_, l_] := Module[{j}, If[n == 0 || i<1, 0, If[l != {} && l[[1]]-i == 1, 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}]]]];
%t a[n_] := g[n, n, {}];
%t Table[a[n], {n, 3, 35}] (* _Jean-François Alcover_, Aug 25 2021, after Maple code *)
%Y Column k=1 of A238707.
%K nonn
%O 3,1
%A _Joerg Arndt_ and _Alois P. Heinz_, Jun 25 2014
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