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%I #9 Aug 28 2021 03:24:50
%S 3,6,35,71,295,751,2326,6524,22309,55992,190282,577410,1951421,
%T 5414977,19405654,64615030,238446543,726141375,2682369977,9475513873,
%U 41043824531,138540753071,524631248766,1902172512592,8404692901429,35025078519164,148160349275671
%N Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 2.
%C Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 2.
%H Alois P. Heinz, <a href="/A244296/b244296.txt">Table of n, a(n) for n = 4..250</a>
%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=2, `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=4..35);
%t h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j +
%t 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 != {} &&
%t l[[1]] - i == 2, If[j = Quotient[n, i]; Mod[n, i] == 0,
%t h[Join[l, Table[i, {j}]]], 0], Sum[g[n - i*j, i - 1,
%t Join [l, Table[i, {j}]]], {j, 0, n/i}]]]];
%t a[n_] := g[n, n, {}];
%t Table[a[n], {n, 4, 35}] (* _Jean-François Alcover_, Aug 28 2021, after Maple code *)
%Y Column k=2 of A238707.
%K nonn
%O 4,1
%A _Joerg Arndt_ and _Alois P. Heinz_, Jun 25 2014
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