cp's OEIS Frontend

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.

%I A217285 #23 Mar 08 2016 08:11:31
%S A217285 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 A217285 8008,11440,12870,11440,8008,4368,1820,560,120,16,1,1,25,300,2300,
%U A217285 12650,53130,177100,480700,1081575,2042975,3268760,4457400,5200300,5200300,4457400,3268760,2042975,1081575,480700,177100,53130,12650,2300,300,25,1
%N 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.
%C A217285 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 A217285 A random labeled relation can be generated in Mathematica:
%C A217285 GraphPlot[g=Table[RandomInteger[],{6},{6}], DirectedEdges->True, VertexLabeling->True, SelfLoopStyle->True, MultiedgeStyle->True]
%C A217285 Sum {k=0...n^2} T(n,k)*k = A185968. - _Geoffrey Critzer_, Oct 07 2012
%H A217285 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 A217285 T(n,k) = binomial(n^2,k).
%F A217285 E.g.f.: Sum{n>=0}(1+y)^(n^2)*x^n/n!. - _Geoffrey Critzer_, Oct 07 2012
%F A217285 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 A217285 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 A217285 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 A217285 Triangle T(n,k) begins:
%e A217285 1;
%e A217285 1,  1;
%e A217285 1,  4,  6,    4,    1;
%e A217285 1,  9,  36,  84,  126,  126,   84,    36,     9,     1;
%e A217285 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, ...
%t A217285 Table[Table[Binomial[n^2,k], {k,0,n^2}], {n,0,6}] //Grid
%o A217285 (PARI) {T(n,k)=polcoeff((1+x+x*O(x^k))^(n^2),k)}
%o A217285 for(n=0,6,for(k=0,n^2,print1(T(n,k),", "));print("")) \\ _Paul D. Hanna_, Aug 22 2013
%o A217285 (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 A217285 {for(n=0,6,for(k=0,n^2,print1(T(n,k),", "));print(""))} \\ _Paul D. Hanna_, Aug 22 2013
%Y A217285 Column k=1 gives: A000290.
%Y A217285 Row lengths are: A002522.
%Y A217285 Antidiagonal sums: A121689.
%K A217285 nonn,tabf
%O A217285 0,5
%A A217285 _Geoffrey Critzer_, Sep 30 2012