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%I #39 Dec 18 2022 12:09:34
%S 1,1,3,9,33,125,531,2349,11205,55589,291423,1583485,8985813,52661609,
%T 319898103,2000390153,12898434825,85374842121,580479540219,
%U 4041838056561,28824970996809,210092964771637,1564766851282299,11890096357039749,92151199272181629
%N Number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.
%C Number of normal semistandard Young tableaux of size n, where a tableau is normal if its entries span an initial interval of positive integers. - _Gus Wiseman_, Feb 23 2018
%H Alois P. Heinz, <a href="/A138178/b138178.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: Sum_{n>=0} Sum_{k=0..n} (-1)^(n-k)*C(n,k)*(1-x)^(-k)*(1-x^2)^(-C(k,2)).
%F G.f.: Sum_{n>=0} 2^(-n-1)*(1-x)^(-n)*(1-x^2)^(-C(n,2)). - _Vladeta Jovovic_, Dec 09 2009
%e a(4) = 33 because there are 1 such matrix of type 1 X 1, 7 matrices of type 2 X 2, 15 of type 3 X 3 and 10 of type 4 X 4, cf. A138177.
%e From _Gus Wiseman_, Feb 23 2018: (Start)
%e The a(3) = 9 normal semistandard Young tableaux:
%e 1 1 2 1 3 1 2 1 1 1 2 3 1 2 2 1 1 2 1 1 1
%e 2 3 2 2 2
%e 3
%e (End)
%e From _Gus Wiseman_, Nov 14 2018: (Start)
%e The a(4) = 33 matrices:
%e [4]
%e .
%e [30][21][20][11][10][02][01]
%e [01][10][02][11][03][20][12]
%e .
%e [200][200][110][101][100][100][100][100][011][010][010][010][001][001][001]
%e [010][001][100][010][020][011][010][001][100][110][101][100][020][010][001]
%e [001][010][001][100][001][010][002][011][100][001][010][002][100][101][110]
%e .
%e [1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
%e [0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
%e [0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
%e [0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
%e (End)
%p gf:= proc(j) local k, n; add(add((-1)^(n-k) *binomial(n, k) *(1-x)^(-k) *(1-x^2)^(-binomial(k, 2)), k=0..n), n=0..j) end: a:= n-> coeftayl(gf(n+1), x=0, n): seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 25 2008
%t Table[Sum[SeriesCoefficient[1/(2^(k+1)*(1-x)^k*(1-x^2)^(k*(k-1)/2)),{x,0,n}],{k,0,Infinity}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 03 2014 *)
%t multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],Sort[Reverse/@#]==#]&]],{n,5}] (* _Gus Wiseman_, Nov 14 2018 *)
%Y Row sums of A138177.
%Y Cf. A007716, A120733, A135588, A296188.
%Y Cf. A057151, A104601, A104602, A120732, A316983, A320796, A321401, A321405, A321407.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Mar 03 2008
%E More terms from _Alois P. Heinz_, Sep 25 2008
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