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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. 2

%I

%S 1,1,1,1,4,6,4,1,1,9,36,84,126,126,84,36,9,1,1,16,120,560,1820,4368,

%T 8008,11440,12870,11440,8008,4368,1820,560,120,16,1,1,25,300,2300,

%U 12650,53130,177100,480700,1081575,2042975,3268760,4457400,5200300,5200300,4457400,3268760,2042975,1081575,480700,177100,53130,12650,2300,300,25,1

%N 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.

%C 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%.

%C A random labeled relation can be generated in Mathematica:

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

%C Sum {k=0...n^2} T(n,k)*k = A185968. - _Geoffrey Critzer_, Oct 07 2012

%H Paul D. Hanna, <a href="/A217285/b217285.txt">Rows 0..20, as a flattened table of n, a(n) for n = 0..2890.</a>

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

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

%F 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

%F 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

%e G.f.: A(x,y) = 1 + x*(1+y) + x^2*(1+y)^4 + x^3*(1+y)^9 + x^4*(1+y)^16 +...

%e Triangle T(n,k) begins:

%e 1;

%e 1, 1;

%e 1, 4, 6, 4, 1;

%e 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;

%e 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, ...

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

%o (PARI) {T(n,k)=polcoeff((1+x+x*O(x^k))^(n^2),k)}

%o for(n=0,6,for(k=0,n^2,print1(T(n,k),", "));print("")) \\ _Paul D. Hanna_, Aug 22 2013

%o (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)}

%o {for(n=0,6,for(k=0,n^2,print1(T(n,k),", "));print(""))} \\ _Paul D. Hanna_, Aug 22 2013

%Y Column k=1 gives: A000290.

%Y Row lengths are: A002522.

%Y Antidiagonal sums: A121689.

%K nonn,tabf

%O 0,5

%A _Geoffrey Critzer_, Sep 30 2012

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Last modified April 16 09:54 EDT 2021. Contains 343034 sequences. (Running on oeis4.)