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A228900 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, (n-k)*k) * y^k ), as read by rows. 6
1, 1, 1, 1, 3, 1, 1, 15, 15, 1, 1, 155, 484, 155, 1, 1, 2685, 36068, 36068, 2685, 1, 1, 65517, 5082340, 15763254, 5082340, 65517, 1, 1, 2063205, 1179126560, 13201421078, 13201421078, 1179126560, 2063205, 1, 1, 79715229, 411708127954, 19954261054442, 61092286569334, 19954261054442, 411708127954, 79715229, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..44.

EXAMPLE

This triangle begins:

1;

1, 1;

1, 3, 1;

1, 15, 15, 1;

1, 155, 484, 155, 1;

1, 2685, 36068, 36068, 2685, 1;

1, 65517, 5082340, 15763254, 5082340, 65517, 1;

1, 2063205, 1179126560, 13201421078, 13201421078, 1179126560, 2063205, 1;

1, 79715229, 411708127954, 19954261054442, 61092286569334, 19954261054442, 411708127954, 79715229, 1;

...

G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+15*y+15*y^2+y^3)*x^3 + (1+155*y+484*y^2+155*y^3+y^4)*x^4 + (1+2685*y+36068*y^2+36068*y^3+2685*y^4+y^5)*x^5 +...

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

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

+ (1 + 4*y + y^2)*x^2/2

+ (1 + 36*y + 36*y^2 + y^3)*x^3/3

+ (1 + 560*y + 1820*y^2 + 560*y^3 + y^4)*x^4/4

+ (1 + 12650*y + 177100*y^2 + 177100*y^3 + 12650*y^4 + y^5)*x^5/5

+ (1 + 376992*y + 30260340*y^2 + 94143280*y^3 + 30260340*y^4 + 376992*y^5 + y^6)*x^6/6 +...

in which the coefficients form A228836(n,k) = binomial(n^2, (n-k)*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)*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. A207135 (row sums), A207137 (antidiagonal sums), A228901 (column 1).

Cf. related triangles: A228836 (log), A209196, A228902, A228904.

Sequence in context: A173917 A174410 A156690 * A060325 A135021 A087987

Adjacent sequences:  A228897 A228898 A228899 * A228901 A228902 A228903

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Sep 07 2013

STATUS

approved

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Last modified October 13 22:28 EDT 2019. Contains 327983 sequences. (Running on oeis4.)