A097190 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 9^n, where R_n(y) forms the initial (n+1) terms of g.f. A097191(y)^(n+1).

1, 1, 24, 1, 36, 612, 1, 48, 1104, 15912, 1, 60, 1740, 32130, 417690, 1, 72, 2520, 56700, 912492, 11027016, 1, 84, 3444, 91350, 1750014, 25562628, 292215924, 1, 96, 4512, 137808, 3059856, 52303968, 710025264, 7764594552, 1, 108, 5724, 197802, 4992354

Offset

0,3

Links

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

Formula

G.f.: A(x, y) = 3*y/((1-27*x*y) + (3*y-1)*(1-27*x*y)^(8/9)).

G.f.: A(x, y) = A097192(x*y)/(1 - x*A097193(x*y)).

Example

Row polynomials evaluated at y=1/3 equals powers of 9:
9^1 = 1 + 24/3;
9^2 = 1 + 36/3 + 612/3^2;
9^3 = 1 + 48/3 + 1104/3^2 + 15912/3^3;
9^4 = 1 + 60/3 + 1740/3^2 + 32130/3^3 + 417690/3^4;
where A097191(y)^(n+1) has the same initial terms as the n-th row:
A097191(y) = 1 + 12y + 60y^2 + 90y^3 - 558y^4 - 2916y^5 + 2160y^6 +...
A097191(y)^2 = 1 + 24y +...
A097191(y)^3 = 1 + 36y + 612y^2 +...
A097191(y)^4 = 1 + 48y + 1104y^2 + 15912y^3 +...
A097191(y)^5 = 1 + 60y + 1740y^2 + 32130y^3 + 417690y^4 +...
Rows begin with n=0:
  1;
  1, 24;
  1, 36,  612;
  1, 48, 1104,  15912;
  1, 60, 1740,  32130,  417690;
  1, 72, 2520,  56700,  912492, 11027016;
  1, 84, 3444,  91350, 1750014, 25562628, 292215924;
  1, 96, 4512, 137808, 3059856, 52303968, 710025264, 7764594552; ...

Mathematica

Table[SeriesCoefficient[3*y/((1-27*x*y) + (3*y-1)*(1-27*x*y)^(8/9)), {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*(9^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. A097186, A097191, A097192, A097193, A097194, A097195.

Sequence in context: A036166 A097013 A040598 * A103903 A280631 A040599

Adjacent sequences: A097187 A097188 A097189 * A097191 A097192 A097193

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 03 2004

Status

approved