A209196 - OEIS

%I #16 Mar 30 2012 18:37:37

%S 1,1,1,1,4,1,1,32,32,1,1,487,3282,487,1,1,11113,657573,657573,11113,1,

%T 1,335745,209282906,1513844855,209282906,335745,1,1,12607257,

%U 96673776804,5580284351032,5580284351032,96673776804,12607257,1,1,565877928,61162554558200

%N Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * y^k ), as read by rows.

%H Paul D. Hanna, <a href="/A209196/b209196.txt">Rows n = 0..30, flattened.</a>

%F Column 1 equals A209197.

%F Row sums equal A167006.

%F Antidiagonal sums equal A206830.

%e This triangle begins:

%e 1;

%e 1, 1;

%e 1, 4, 1;

%e 1, 32, 32, 1;

%e 1, 487, 3282, 487, 1;

%e 1, 11113, 657573, 657573, 11113, 1;

%e 1, 335745, 209282906, 1513844855, 209282906, 335745, 1;

%e 1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1;

%e 1, 565877928, 61162554558200, 31336815578461815, 229089181252258800, 31336815578461815, 61162554558200, 565877928, 1; ...

%e G.f.: A(x,y) = 1 + (1+y)*x + (1+4*y+y^2)*x^2 + (1+32*y+32*y^2+y^3)*x^3 + (1+487*y+3282*y^2+487*y^3+y^4)*x^4 +...

%e The logarithm of the g.f. equals the series:

%e log(A(x,y)) = (1 + y)*x

%e + (1 + 6*y + y^2)*x^2/2

%e + (1 + 84*y + 84*y^2 + y^3)*x^3/3

%e + (1 + 1820*y + 12870*y^2 + 1820*y^3 + y^4)*x^4/4

%e + (1 + 53130*y + 3268760*y^2 + 3268760*y^3 + 53130*y^4 + y^5)*x^5/5 +...

%e in which the coefficients form A209330(n,k) = binomial(n^2, n*k).

%o (PARI) {T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,x^m/m*sum(j=0,m,binomial(m^2,m*j)*y^j))+x*O(x^n)),n,x),k,y)}

%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

%Y Cf. A209197, A167006, A206830, A209330 (log), A155200.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Mar 05 2012