editing

approved

1, 2, 1, 10, 6, 1, 188, 82, 14, 1, 16774, 4452, 490, 30, 1, 6745436, 1074934, 71108, 2602, 62, 1, 11466849412, 1082704500, 43173414, 951300, 13002, 126, 1, 80444398636280, 4411700155252, 104251164804, 1387446246, 11470404, 62538, 254, 1, 2306003967992402758, 72146891831948808, 989785148972932, 7803708940836, 38993810694, 129076164, 292810, 510, 1, 268654794629082985019564, 4724816968764733073446, 36967624172237518088, 169140002768370820, 500466007443108, 1001353593606, 1382564804, 1343434, 1022, 1

2306003967992402758, 72146891831948808, 989785148972932, 7803708940836, 38993810694, 129076164, 292810, 510, 1; ...

268654794629082985019564, 4724816968764733073446, 36967624172237518088, 169140002768370820, 500466007443108, 1001353593606, 1382564804, 1343434, 1022, 1; ...

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

approved

editing

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Feb 04 2009

Triangle, read by rows, where g.f.: A(x,y) = exp( Sum_{n>=1} (2^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

1, 2, 1, 10, 6, 1, 188, 82, 14, 1, 16774, 4452, 490, 30, 1, 6745436, 1074934, 71108, 2602, 62, 1, 11466849412, 1082704500, 43173414, 951300, 13002, 126, 1, 80444398636280, 4411700155252, 104251164804, 1387446246, 11470404, 62538

0,2

More generally, for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

G.f.: A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k.

G.f.: A(x,y) = 1 + (2 + y)x + (10 + 6y + y^2)x^2 + (188 + 82y + 14y^2 + y^3)x^3 +...

Triangle begins:

1;

2, 1;

10, 6, 1;

188, 82, 14, 1;

16774, 4452, 490, 30, 1;

6745436, 1074934, 71108, 2602, 62, 1;

11466849412, 1082704500, 43173414, 951300, 13002, 126, 1;

80444398636280, 4411700155252, 104251164804, 1387446246, 11470404, 62538, 254, 1;

2306003967992402758, 72146891831948808, 989785148972932, 7803708940836, 38993810694, 129076164, 292810, 510, 1; ...

(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n+1, (2^m+y)^m*x^m/m)+x*O(x^n)), n, x), k, y)}

Cf. A155200 (column 0), A155201 (row sums), A155811 (column 1).

nonn,tabl,new

Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2009

approved