A097186 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 3^n, where R_n(y) forms the initial (n+1) terms of g.f. A057083(y)^(n+1).

1, 1, 6, 1, 9, 45, 1, 12, 78, 360, 1, 15, 120, 675, 2970, 1, 18, 171, 1134, 5859, 24948, 1, 21, 231, 1764, 10458, 51030, 212058, 1, 24, 300, 2592, 17334, 95256, 445824, 1817640, 1, 27, 378, 3645, 27135, 165726, 861597, 3905253, 15677145, 1, 30, 465, 4950, 40590, 272646, 1557765, 7760610, 34285680, 135868590

Offset

0,3

Comments

Row sums form A097189. Main diagonal is A004988. Ratio of g.f.s of any two adjacent diagonals equals g.f. of A097188, where the g.f.s satisfy: A057083(x*A097188(x)) = A097188(x).

Links

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Formula

G.f.: A(x, y) = 3*y/((1-9*x*y) + (3*y-1)*(1-9*x*y)^(2/3)).

G.f.: A(x, y) = A004988(x*y)/(1 - x*A097188(x*y)).

Example

Row polynomials evaluated at y=1/3 equals powers of 3:
3^1 = 1 + 6/3;
3^2 = 1 + 9/3 + 45/3^2;
3^3 = 1 + 12/3 + 78/3^2 + 360/3^3;
3^4 = 1 + 15/3 + 120/3^2 + 675/3^3 + 2970/3^4;
where A057083(y)^(n+1) has the same initial terms as the n-th row:
A057083(y) = 1 + 3y + 6y^2 + 9y^3 + 9y^4 + 0y^5 - 27y^6 +...
A057083(y)^2 = 1 + 6y +...
A057083(y)^3 = 1 + 9y + 45y^2 +...
A057083(y)^4 = 1 + 12y + 78y^2 + 360y^3 +...
A057083(y)^5 = 1 + 15y + 120y^2 + 675y^3 + 2970y^4 +...
Rows begin with n=0:
  1;
  1,  6;
  1,  9,  45;
  1, 12,  78,  360;
  1, 15, 120,  675,  2970;
  1, 18, 171, 1134,  5859,  24948;
  1, 21, 231, 1764, 10458,  51030, 212058;
  1, 24, 300, 2592, 17334,  95256, 445824, 1817640;
  1, 27, 378, 3645, 27135, 165726, 861597, 3905253, 15677145; ...

Mathematica

Table[SeriesCoefficient[3y/((1-9xy) - (1-3y)*(1-9xy)^(2/3)), {x, 0, n}, {y, 0, k}], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 17 2019 *)

Prog

(PARI) {T(n, k)=if(n==0, 1, if(k==0, 1, if(k==n, 3^n*(3^n -sum(j=0, n-1, T(n, j)/3^j)), polcoeff((Ser(vector(n, i, T(n-1, i-1)), x) +x*O(x^k))^((n+1)/n), k, x))))}

Crossrefs

Cf. A004988, A057083, A097188, A097189, A097190.

Sequence in context: A340041 A073446 A097179 * A329047 A363316 A347130

Adjacent sequences: A097183 A097184 A097185 * A097187 A097188 A097189

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 03 2004

Extensions

More terms added by G. C. Greubel, Sep 17 2019

Status

approved