login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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(Sum((-1)^(n-k)*C(n,k)*(1-x)^(-k)*(1-x^2)^(-C(k,2)), k=0..n), n=0..infinity).

G.f.: Sum(2^(-n-1)*(1-x)^(-n)*(1-x^2)^(-C(n,2)),n=0..infinity). - Vladeta Jovovic, Dec 09 2009

EXAMPLE

a(4) = 33 because there are 1 such matrix of type 1x1, 7 matrices of type 2 X 2, 15 of type 3 X 3 and 10 of type 4 X 4, cf. A138177.

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 - Gus Wiseman, Feb 23 2018

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 18 01:05 EST 2020. Contains 330995 sequences. (Running on oeis4.)