A209196 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.

1, 1, 1, 1, 4, 1, 1, 32, 32, 1, 1, 487, 3282, 487, 1, 1, 11113, 657573, 657573, 11113, 1, 1, 335745, 209282906, 1513844855, 209282906, 335745, 1, 1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1, 1, 565877928, 61162554558200

Offset: 0

Author

Paul D. Hanna, Mar 05 2012

Keywords

• nonn
• tabl

Examples

			This triangle begins:
1;
1, 1;
1, 4, 1;
1, 32, 32, 1;
1, 487, 3282, 487, 1;
1, 11113, 657573, 657573, 11113, 1;
1, 335745, 209282906, 1513844855, 209282906, 335745, 1;
1, 12607257, 96673776804, 5580284351032, 5580284351032, 96673776804, 12607257, 1;
1, 565877928, 61162554558200, 31336815578461815, 229089181252258800, 31336815578461815, 61162554558200, 565877928, 1; ...
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 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 6*y + y^2)*x^2/2
+ (1 + 84*y + 84*y^2 + y^3)*x^3/3
+ (1 + 1820*y + 12870*y^2 + 1820*y^3 + y^4)*x^4/4
+ (1 + 53130*y + 3268760*y^2 + 3268760*y^3 + 53130*y^4 + y^5)*x^5/5 +...
in which the coefficients form A209330(n,k) = binomial(n^2, n*k).
		

Links

• Paul D. Hanna, Rows n = 0..30, flattened.

Crossrefs

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

Programs

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

Formula

Column 1 equals A209197.

Row sums equal A167006.

Antidiagonal sums equal A206830.