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A138177 Triangle T(n,k) read by rows: number of k X k symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n, n>=1, 1<=k<=n. 6
1, 1, 2, 1, 4, 4, 1, 7, 15, 10, 1, 10, 36, 52, 26, 1, 14, 74, 176, 190, 76, 1, 18, 132, 460, 810, 696, 232, 1, 23, 222, 1060, 2705, 3756, 2674, 764, 1, 28, 347, 2180, 7565, 15106, 17262, 10480, 2620, 1, 34, 525, 4204, 19013, 51162, 83440, 80816, 42732, 9496, 1, 40 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

See the Brualdi/Ma reference for the connection to A161126. - Joerg Arndt, Nov 02 2014

T(n,k) is also the number of semistandard Young tableaux of size n whose entries span the interval 1..k. See also Gus Wiseman's comment in A138178. The T(4,2) = 7 semi-standard Young tableaux of size 4 spanning the interval 1..2 are:

   11  122  112  111  1222  1122  1112

   22  2    2    2                      . - Jacob Post, Jun 15 2018

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Richard A. Brualdi, Shi-Mei Ma, Enumeration of involutions by descents and symmetric matrices, European Journal of Combinatorics, vol.43, pp.220-228, (January 2015).

FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Samantha Dahlberg, Combinatorial Proofs of Identities Involving Symmetric Matrices, arXiv:1410.7356 [math.CO], (27-October-2014)

FORMULA

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A210391(n,k-i). - Alois P. Heinz, Apr 06 2015

EXAMPLE

Triangle T(n,k) begins:

  1;

  1,  2;

  1,  4,   4;

  1,  7,  15,   10;

  1, 10,  36,   52,   26;

  1, 14,  74,  176,  190,   76;

  1, 18, 132,  460,  810,  696,  232;

  1, 23, 222, 1060, 2705, 3756, 2674, 764;

  ...

MAPLE

gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):

A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):

T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):

seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Apr 06 2015

MATHEMATICA

gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); A[n_, k_] := Coefficient[ Series [gf[k], {x, 0, n+1}], x, n]; T[n_, k_] := Sum[(-1)^j*Binomial[k, j]*A[n, k-j], {j, 0, k}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

CROSSREFS

Cf. (row sums) A138178, A135589, A135588, A161126, A210391.

Main diagonal gives A000085. - Alois P. Heinz, Apr 06 2015

T(2n,n) gives A266305.

T(n^2,n) gives A268309.

Sequence in context: A214984 A118976 A210235 * A101559 A220537 A229717

Adjacent sequences:  A138174 A138175 A138176 * A138178 A138179 A138180

KEYWORD

nonn,tabl

AUTHOR

Vladeta Jovovic, Mar 03 2008

STATUS

approved

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Last modified December 17 02:28 EST 2018. Contains 318192 sequences. (Running on oeis4.)