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A228902 Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) * y^k ), as read by rows. 6
1, 1, 1, 1, 3, 1, 1, 6, 45, 1, 1, 10, 505, 2905, 1, 1, 15, 3045, 412044, 411500, 1, 1, 21, 12880, 16106168, 1218805926, 100545716, 1, 1, 28, 43176, 309616264, 479536629727, 9030648908720, 37614371968, 1, 1, 36, 122640, 3752248896, 61545730104024, 50139332516318674, 139855355007409180, 19977489354808, 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, 6, 45, 1;

1, 10, 505, 2905, 1;

1, 15, 3045, 412044, 411500, 1;

1, 21, 12880, 16106168, 1218805926, 100545716, 1;

1, 28, 43176, 309616264, 479536629727, 9030648908720, 37614371968, 1;

1, 36, 122640, 3752248896, 61545730104024, 50139332516318674, 139855355007409180, 19977489354808, 1;

...

G.f.: A(x,y) = 1 + (1+y)*x + (1+3*y+y^2)*x^2 + (1+6*y+45*y^2+y^3)*x^3 + (1+10*y+505*y^2+2905*y^3+y^4)*x^4 + (1+15*y+3045*y^2+412044*y^3+411500*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 + 9*y + 126*y^2 + y^3)*x^3/3

+ (1 + 16*y + 1820*y^2 + 11440*y^3 + y^4)*x^4/4

+ (1 + 25*y + 12650*y^2 + 2042975*y^3 + 2042975*y^4 + y^5)*x^5/5

+ (1 + 36*y + 58905*y^2 + 94143280*y^3 + 7307872110*y^4 + 600805296*y^5 + y^6)*x^/6 +...

in which the coefficients form A226234(n,k) = binomial(n^2, k^2).

PROG

(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m^2, 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. A206848 (row sums), A206850 (antidiagonal sums), A228903 (diagonal).

Cf. related triangles:  A226234 (log), A209196, A228900, A228904.

Sequence in context: A228899 A255918 A102479 * A053193 A010273 A046143

Adjacent sequences:  A228899 A228900 A228901 * A228903 A228904 A228905

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Sep 07 2013

STATUS

approved

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Last modified February 28 10:22 EST 2020. Contains 332323 sequences. (Running on oeis4.)