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 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. 45
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 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 EXAMPLE 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) 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 *) CROSSREFS 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 KEYWORD easy,nonn AUTHOR Vladeta Jovovic, Mar 03 2008 EXTENSIONS More terms from Alois P. Heinz, Sep 25 2008 STATUS approved

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Last modified March 28 14:50 EDT 2023. Contains 361595 sequences. (Running on oeis4.)