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%I #26 Feb 17 2017 02:33:39
%S 1,1,1,2,3,5,16,49,70,168,768,3300,7887,15015,48048,292864,1946516,
%T 4934930,14454726,34918884,141892608,1100742656,9732668946,
%U 32773404950,97848532782,344699731090,1020872973120,5091106775040,48608795688960,586393249199550
%N Number of standard Young tableaux for partitions of n into distinct parts with largest part floor(sqrt(2*n)+1/2).
%C a(n) is the leftmost nonzero element in row n of A219272, A219274.
%C Floor(sqrt(2*n)+1/2) = A002024(n) for n>0. There are no partitions of n into distinct parts with a smaller largest part.
%H Alois P. Heinz, <a href="/A219339/b219339.txt">Table of n, a(n) for n = 0..300</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F a(n) = A219272(n,floor(sqrt(2*n)+1/2)) = A219274(n,floor(sqrt(2*n)+1/2)).
%e For n=5, we have floor(sqrt(2*n)+1/2) = 3, and a(5) = 5, because there are 5 standard Young tableaux for partitions of 5 into distinct parts with largest part 3:
%e +---------+ +---------+ +---------+ +---------+ +---------+
%e | 1 2 3 | | 1 2 4 | | 1 2 5 | | 1 3 4 | | 1 3 5 |
%e | 4 5 .--+ | 3 5 .--+ | 3 4 .--+ | 2 5 .--+ | 2 4 .--+
%e +------+ +------+ +------+ +------+ +------+
%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 s; s:=i*(i+1)/2;
%p `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
%p g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
%p end:
%p a:= n-> g(n, floor(sqrt(2*n)+1/2), []):
%p seq(a(n), n=0..30);
%t h[l_] := (n = Length[l]; Total[l]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]); g[n_, i_, l_] := g[n, i, l] = (s = i*(i+1)/2; If[n==s, h[Join[l, Table[i-j, {j, 0, i-1}]] ], If[n>s, 0, g[n, i-1, l]+If[i>n, 0, g[n-i, i-1, Append[l, i]]]]] ); a[n_] := g[n, Floor[Sqrt[2*n]+1/2], {}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 16 2017, translated from Maple *)
%Y Cf. A005118 (subsequence), A219347.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Nov 18 2012
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