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A138178
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Number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.
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45
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..500
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FORMULA
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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
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EXAMPLE
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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]
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[30][21][20][11][10][02][01]
[01][10][02][11][03][20][12]
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[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]
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[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)
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Row sums of A138177.
Cf. A007716, A120733, A135588, A296188.
Cf. A057151, A104601, A104602, A120732, A316983, A320796, A321401, A321405, A321407.
Sequence in context: A049171 A050387 A049157 * A063027 A148998 A049185
Adjacent sequences: A138175 A138176 A138177 * A138179 A138180 A138181
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KEYWORD
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easy,nonn
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AUTHOR
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Vladeta Jovovic, Mar 03 2008
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EXTENSIONS
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More terms from Alois P. Heinz, Sep 25 2008
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STATUS
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approved
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