A097179 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 4^n, where R_n(y) forms the initial (n+1) terms of g.f. A077860(y)^(n+1).

1, 1, 6, 1, 9, 42, 1, 12, 74, 308, 1, 15, 115, 595, 2310, 1, 18, 165, 1020, 4746, 17556, 1, 21, 224, 1610, 8722, 37730, 134596, 1, 24, 292, 2392, 14778, 73080, 299508, 1038312, 1, 27, 369, 3393, 23535, 130851, 604707, 2376099, 8046918

Offset

0,3

Comments

Row sums form A097180. Diagonal is A004982. Ratio of g.f.s of any two adjacent diagonals equals g.f. of A048779, where the g.f.s satisfy: A077860(x*A048779(x)) = A048779(x).

Links

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

Formula

G.f.: A(x, y) = 2*y/((1-8*x*y) + (2*y-1)*(1-8*x*y)^(3/4)).

G.f.: A(x, y) = A004982(x*y)/(1 - x*A048779(x*y)).

Example

Row polynomials evaluated at y=1/2 equals powers of 4:
4^1 = 1 + 6/2;
4^2 = 1 + 9/2 + 42/2^2;
4^3 = 1 + 12/2 + 74/2^2 + 308/2^3;
4^4 = 1 + 15/2 + 115/2^2 + 595/2^3 + 2310/2^4;
where A077860(y)^(n+1) has the same initial terms as the n-th row:
A077860(y) = 1 +3*y +5*y^2 +5*y^3 +1*y^4 -7*y^5 -15*y^6 -15*y^7 +...
A077860(y)^2 = 1 + 6*y +...
A077860(y)^3 = 1 + 9*y + 42*y^2 +...
A077860(y)^4 = 1 + 12*y + 74*y^2 + 308*y^3 +...
A077860(y)^5 = 1 + 15*y + 115*y^2 + 595*y^3 + 2310*y^4 +...
Rows begin with n=0:
  1;
  1,  6;
  1,  9,  42;
  1, 12,  74,  308;
  1, 15, 115,  595,  2310;
  1, 18, 165, 1020,  4746, 17556;
  1, 21, 224, 1610,  8722, 37730,  134596;
  1, 24, 292, 2392, 14778, 73080,  299508, 1038312;
  1, 27, 369, 3393, 23535, 130851, 604707, 2376099, 8046918; ...

Mathematica

Table[SeriesCoefficient[2*y/((1-8*x*y) +(2*y-1)*(1-8*x*y)^(3/4)), {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. A004982, A048779, A077860, A097180, A097181.

Sequence in context: A364618 A340041 A073446 * A097186 A329047 A363316

Adjacent sequences: A097176 A097177 A097178 * A097180 A097181 A097182

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 02 2004

Status

approved