A097181 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 8^n, where R_n(y) forms the initial (n+1) terms of g.f. A097182(y)^(n+1).

1, 1, 14, 1, 21, 210, 1, 28, 378, 3220, 1, 35, 595, 6475, 49910, 1, 42, 861, 11396, 108402, 778596, 1, 49, 1176, 18326, 207074, 1791930, 12198004, 1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920, 1, 63, 1953, 39585, 587727, 6783147

Offset

0,3

Comments

Row sums form A097185. Diagonal is A097183.

Ratio of g.f.s of any two adjacent diagonals equals g.f. of A097184, where the g.f.s satisfy: A097182(x*A097184(x)) = A097184(x).

Links

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

Formula

G.f.: A(x, y) = 2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)).

G.f.: A(x, y) = A097183(x*y)/(1 - x*A097184(x*y)).

Example

Row polynomials evaluated at y=1/2 equals powers of 8:
8^1 = 1 + 14/2;
8^2 = 1 + 21/2 + 210/2^2;
8^3 = 1 + 28/2 + 378/2^2 + 3220/2^3;
8^4 = 1 + 35/2 + 595/2^2 + 6475/2^3 + 49910/2^4;
where A097182(y)^(n+1) has the same initial terms as the n-th row:
A097182(y) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 -+...
A097182(y)^2 = 1 + 14y +...
A097182(y)^3 = 1 + 21y + 210y^2 +...
A097182(y)^4 = 1 + 28y + 378y^2 + 3220y^3 +...
A097182(y)^5 = 1 + 35y + 595y^2 + 6475y^3 + 49910y^4 +...
Rows begin with n=0:
  1;
  1, 14;
  1, 21,  210;
  1, 28,  378,  3220;
  1, 35,  595,  6475,  49910;
  1, 42,  861, 11396, 108402,  778596;
  1, 49, 1176, 18326, 207074, 1791930, 12198004;
  1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920;
  1, 63, 1953, 39585, 587727, 6783147, 62974371, 479497491, 3019005990; ...

Mathematica

Table[SeriesCoefficient[2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)), {x, 0, n}, {y, 0, k}], {n, 0, 10}, {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, 2^n*(4^n -sum(j=0, n-1, T(n, j)/2^j)), polcoeff((Ser(vector(n, i, T(n-1, i-1)), x) +x*O(x^k))^((n+1)/n), k, x))))}

Crossrefs

Cf. A097179, A097182, A097183, A097184, A097185, A097186.

Sequence in context: A040207 A036163 A040208 * A040209 A071713 A147716

Adjacent sequences: A097178 A097179 A097180 * A097182 A097183 A097184

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 03 2004

Status

approved