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A228904 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) * y^k ), as read by rows. 7
1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 26, 62, 1, 1, 5, 70, 1087, 1031, 1, 1, 6, 155, 9257, 124702, 24782, 1, 1, 7, 301, 51397, 4479983, 26375325, 774180, 1, 1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1, 1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..54.

EXAMPLE

This triangle begins:

1;

1, 1;

1, 2, 1;

1, 3, 7, 1;

1, 4, 26, 62, 1;

1, 5, 70, 1087, 1031, 1;

1, 6, 155, 9257, 124702, 24782, 1;

1, 7, 301, 51397, 4479983, 26375325, 774180, 1;

1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1;

1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1;

...

G.f.: A(x,y) = 1 + (1+y)*x + (1+2*y+y^2)*x^2 + (1+3*y+7*y^2+y^3)*x^3 + (1+4*y+26*y^2+62*y^3+y^4)*x^4 + (1+5*y+70*y^2+1087*y^3+1031*y^4+y^5)*x^5 +...

The logarithm of the g.f. equals the series:

log(A(x,y)) = (1 + y)*x

+ (1 + 2*y + y^2)*x^2/2

+ (1 + 3*y + 15*y^2 + y^3)*x^3/3

+ (1 + 4*y + 70*y^2 + 220*y^3 + y^4)*x^4/4

+ (1 + 5*y + 210*y^2 + 5005*y^3 + 4845*y^4 + y^5)*x^5/5

+ (1 + 6*y + 495*y^2 + 48620*y^3 + 735471*y^4 + 142506*y^5 + y^6)*x^6/6 +...

in which the coefficients form A228832(n,k) = binomial(n*k, k^2).

PROG

(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m*j, j^2)*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. A228809 (row sums), A228905 (antidiagonal sums), A228906 (diagonal).

Cf. related triangles: A228832 (log), A209196, A228900, A228902.

Sequence in context: A025243 A341014 A145085 * A144512 A159314 A135701

Adjacent sequences:  A228901 A228902 A228903 * A228905 A228906 A228907

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Sep 07 2013

STATUS

approved

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Last modified October 25 11:12 EDT 2021. Contains 348246 sequences. (Running on oeis4.)