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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 12 21:19 EDT 2021. Contains 344967 sequences. (Running on oeis4.)