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%I #19 Jul 02 2015 05:46:43
%S 1,1,1,1,4,5,15,21,56,246,525,1573,5764,14092,41405,136995,772552,
%T 2148290,8806629,31679365,155743665,495240074,2049655762,7403470138,
%U 32627363920,207316068370,784695179515,3721285661481,16967347935561,82192321793926,455572563875425
%N Number of standard Young tableaux with shapes corresponding to partitions into distinct parts with minimal difference 2.
%H Alois P. Heinz, <a href="/A225121/b225121.txt">Table of n, a(n) for n = 0..120</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_tableau">Young tableau</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 s; s:=ceil(i*(i+2)/4);
%p `if`(n=s, h([l[], seq(i-2*j, j=0..iquo(i-1,2))]), `if`(n>s, 0,
%p g(n, i-1, l)+`if`(i>n, 0, g(n-i, i-2, [l[], i]))))
%p end:
%p a:= n-> g(n, n, []):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Apr 29 2013
%t h[l_List] := Module[{n}, 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_List] := Module[{s}, s = Ceiling[i*(i+2)/4]; If[n==s, h[Join[l, Table[i-2*j, {j, 0, Quotient[i-1, 2]}]]], If[n>s, 0, g[n, i-1, l] + If[i>n, 0, g[n-i, i-2, Append[l, i]]]]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jul 02 2015, after _Alois P. Heinz_ *)
%Y Cf. A218293 (tableaux with shapes corresponding to partitions into distinct parts).
%Y Cf. A000085 (standard Young tableaux for all shapes).
%K nonn
%O 0,5
%A _Joerg Arndt_, Apr 29 2013
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