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

0,5

Links

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

Formula

Column 1 equals A209197.

Row sums equal A167006.

Antidiagonal sums equal A206830.

Example

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).

Prog

(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(""))

Crossrefs

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

Sequence in context: A173918 A174412 A177939 * A158390 A228836 A176419

Adjacent sequences: A209193 A209194 A209195 * A209197 A209198 A209199

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 05 2012

Status

approved