A209424 Triangle defined by g.f.: A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n, k)^n * y^k ), as read by rows.

1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 76, 347, 76, 1, 1, 701, 20429, 20429, 701, 1, 1, 8477, 1919660, 10707908, 1919660, 8477, 1, 1, 126126, 259227625, 9203978774, 9203978774, 259227625, 126126, 1, 1, 2223278, 47484618291, 12099129236936, 72078431500368

Offset: 0

Author

Paul D. Hanna, Mar 08 2012

Keywords

• nonn
• tabl

Comments

Column 1 is A060946.

Column 2 is A209425.

Row sums equal A167007.

Antidiagonal sums equal A166894.

Central terms form A209426.

Examples

			This triangle begins:
1;
1, 1;
1, 3, 1;
1, 12, 12, 1;
1, 76, 347, 76, 1;
1, 701, 20429, 20429, 701, 1;
1, 8477, 1919660, 10707908, 1919660, 8477, 1;
1, 126126, 259227625, 9203978774, 9203978774, 259227625, 126126, 1;
1, 2223278, 47484618291, 12099129236936, 72078431500368, 12099129236936, 47484618291, 2223278, 1; ...
G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+12*y+12*y^2+y^3)*x^3 + (1+76*y+20429*y^2+76*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^2*y + y^2)*x^2/2
+ (1 + 3^3*y + 3^3*y^2 + y^3)*x^3/3
+ (1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4)*x^4/4
+ (1 + 5^5*y + 10^5*y^2 + 10^5*y^3 + 5^5*y^4 + y^5)*x^5/5 +...
in which the coefficients are found in triangle A209427.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..495 for Rows 0..30 of this triangle in flattened form.

Crossrefs

Cf. A060946, A209425, A167007, A166894, A209426, A209427, A209196.

Programs

PARI

{T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m,k)^m*y^k))+x*O(x^n)),n,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))