A138178 Number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

1, 1, 3, 9, 33, 125, 531, 2349, 11205, 55589, 291423, 1583485, 8985813, 52661609, 319898103, 2000390153, 12898434825, 85374842121, 580479540219, 4041838056561, 28824970996809, 210092964771637, 1564766851282299, 11890096357039749, 92151199272181629

Offset: 0

Author

Vladeta Jovovic, Mar 03 2008

Keywords

• easy
• nonn

Comments

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

Examples

			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.
From _Gus Wiseman_, Feb 23 2018: (Start)
The a(3) = 9 normal semistandard Young tableaux:
1   1 2   1 3   1 2   1 1   1 2 3   1 2 2   1 1 2   1 1 1
2   3     2     2     2
3
(End)
From _Gus Wiseman_, Nov 14 2018: (Start)
The a(4) = 33 matrices:
[4]
.
[30][21][20][11][10][02][01]
[01][10][02][11][03][20][12]
.
[200][200][110][101][100][100][100][100][011][010][010][010][001][001][001]
[010][001][100][010][020][011][010][001][100][110][101][100][020][010][001]
[001][010][001][100][001][010][002][011][100][001][010][002][100][101][110]
.
[1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
[0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
[0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
[0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Row sums of A138177.

Cf. A007716, A120733, A135588, A296188.

Cf. A057151, A104601, A104602, A120732, A316983, A320796, A321401, A321405, A321407.

Programs

Maple

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

Mathematica

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 *)
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 *)

Formula

G.f.: Sum_{n>=0} Sum_{k=0..n} (-1)^(n-k)*C(n,k)*(1-x)^(-k)*(1-x^2)^(-C(k,2)).

G.f.: Sum_{n>=0} 2^(-n-1)*(1-x)^(-n)*(1-x^2)^(-C(n,2)). - Vladeta Jovovic, Dec 09 2009

Extensions

More terms from Alois P. Heinz, Sep 25 2008