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A244297 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 3.

%I A244297 #15 Aug 28 2021 03:24:42
%S A244297 4,10,69,195,929,3044,11824,40985,158079,539876,2065087,7272937,
%T A244297 27923757,101194930,381940222,1429135919,5607176733,21323561733,
%U A244297 84260636527,325309822037,1337034045619,5421586411034,22509005469068,92412147570641,390023528935516
%N A244297 Number of standard Young tableaux with n cells such that the lengths of the first and the last row differ by 3.
%C A244297 Also the number of ballot sequences of length n such that the multiplicities of the largest and the smallest value differ by 3.
%H A244297 Alois P. Heinz, <a href="/A244297/b244297.txt">Table of n, a(n) for n = 5..150</a>
%p A244297 h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A244297     add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
%p A244297 g:= proc(n, i, l) local j; `if`(n=0 or i<1, 0, `if`(l<>[] and
%p A244297       l[1]-i=3, `if`(irem(n, i, 'j')=0, h([l[], i$j]), 0),
%p A244297       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
%p A244297     end:
%p A244297 a:= n-> g(n$2, []):
%p A244297 seq(a(n), n=5..35);
%t A244297 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}]];
%t A244297 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}]]]];
%t A244297 a[n_] := g[n, n, {}];
%t A244297 Table[a[n], {n, 5, 35}] (* _Jean-François Alcover_, Aug 28 2021, after Maple code *)
%Y A244297 Column k=3 of A238707.
%K A244297 nonn
%O A244297 5,1
%A A244297 _Joerg Arndt_ and _Alois P. Heinz_, Jun 25 2014