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.

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

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 A348649 A010273

Adjacent sequences: A228899 A228900 A228901 * A228903 A228904 A228905

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 07 2013

Status

approved