A217285 Irregular triangle read by rows: T(n,k) is the number of labeled relations on n nodes with exactly k edges; n>=0, 0<=k<=n^2.

1, 1, 1, 1, 4, 6, 4, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 1, 25, 300, 2300, 12650, 53130, 177100, 480700, 1081575, 2042975, 3268760, 4457400, 5200300, 5200300, 4457400, 3268760, 2042975, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1

Offset: 0

Author

Geoffrey Critzer, Sep 30 2012

Keywords

• nonn
• tabf

Comments

A labeled relation on 6 nodes will be connected with probability > 99%. It will have at least 10 and no more than 26 edges with probability > 99%.

A random labeled relation can be generated in Mathematica:

GraphPlot[g=Table[RandomInteger[],{6},{6}], DirectedEdges->True, VertexLabeling->True, SelfLoopStyle->True, MultiedgeStyle->True]

Sum {k=0...n^2} T(n,k)*k = A185968. - Geoffrey Critzer, Oct 07 2012

Examples

			G.f.: A(x,y) = 1 + x*(1+y) + x^2*(1+y)^4 + x^3*(1+y)^9 + x^4*(1+y)^16 +...
Triangle T(n,k) begins:
1;
1,  1;
1,  4,  6,    4,    1;
1,  9,  36,  84,  126,  126,   84,    36,     9,     1;
1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, ...
		

Links

• Paul D. Hanna, Rows 0..20, as a flattened table of n, a(n) for n = 0..2890.

Crossrefs

Column k=1 gives: A000290.

Row lengths are: A002522.

Antidiagonal sums: A121689.

Programs

Mathematica

Table[Table[Binomial[n^2,k], {k,0,n^2}], {n,0,6}] //Grid

PARI

{T(n,k)=polcoeff((1+x+x*O(x^k))^(n^2),k)}
for(n=0,6,for(k=0,n^2,print1(T(n,k),", "));print("")) \\ Paul D. Hanna, Aug 22 2013

PARI

{T(n,k)=polcoeff(polcoeff(sum(m=0, n, x^m*(1+y)^m*prod(k=1, m, (1-x*(1+y)^(4*k-3))/(1-x*(1+y)^(4*k-1) +x*O(x^n)))), n,x),k,y)}
{for(n=0,6,for(k=0,n^2,print1(T(n,k),", "));print(""))} \\ Paul D. Hanna, Aug 22 2013

Formula

T(n,k) = binomial(n^2,k).

E.g.f.: Sum{n>=0}(1+y)^(n^2)*x^n/n!. - Geoffrey Critzer, Oct 07 2012

G.f.: A(x,y) = Sum_{n>=0} x^n*(1+y)^n*Product_{k=1..n} (1-x*(1+y)^(4*k-3))/(1-x*(1+y)^(4*k-1)) due to a q-series identity. - Paul D. Hanna, Aug 22 2013

G.f.: A(x,y) = 1/(1- q*x/(1- (q^3-q)*x/(1- q^5*x/(1- (q^7-q^3)*x/(1- q^9*x/(1- (q^11-q^5)*x/(1- q^13*x/(1- (q^15-q^7)*x/(1- ...))))))))), a continued fraction where q = (1+y), due to an identity of a partial elliptic theta function. - Paul D. Hanna, Aug 22 2013