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

0,5

Comments

Column 1 is A060946.

Column 2 is A209425.

Row sums equal A167007.

Antidiagonal sums equal A166894.

Central terms form A209426.

Links

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

Example

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.

Prog

(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(""))

Crossrefs

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

Sequence in context: A078122 A128592 A156584 * A129619 A094573 A055154

Adjacent sequences: A209421 A209422 A209423 * A209425 A209426 A209427

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 08 2012

Status

approved