A097182 G.f. A(x) has the property that the first (n+1) terms of A(x)^(n+1) form the n-th row polynomial R_n(y) of triangle A097181 and satisfy R_n(1/2) = 8^n for all n>=0.

1, 7, 21, 21, -63, -231, -15, 1521, 3073, -4319, -29631, -29631, 143361, 489345, -255, -3342591, -6684671, 9454081, 64553985, 64553985, -311689215, -1064175615, -4095, 7266627585, 14533263361, -20553129983, -140345589759, -140345589759, 677648531457, 2313636773889

Offset: 0

Author

Paul D. Hanna, Aug 03 2004

Keywords

• sign

Examples

			A(x) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 - 15*x^6 +-...
For n>=0, the first (n+1) coefficients of A(x)^(n+1) forms the
n-th row polynomial R_n(y) of triangle A097181:
A^1 = {1, _7,  21,    21,    -63,    -231,      -15,     1521, ...}
A^2 = {1, 14, _91,   336,    609,    -462,    -5469,    -9516, ...}
A^3 = {1, 21, 210, _1288,   5103,   11655,     2160,   -85590, ...}
A^4 = {1, 28, 378,  3220, _18907,   77280,   199860,   153000, ...}
A^5 = {1, 35, 595,  6475,  49910, _283192,  1175190,  3282870, ...}
A^6 = {1, 42, 861, 11396, 108402,  778596, _4296034, 17959968, ...}
These row polynomials satisfy: R_n(1/2) = 8^n:
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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A097181, A097183, A097184, A097185.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 16*x/(1-(1-2*x)^8) )); // G. C. Greubel, Sep 17 2019

Maple

seq(coeff(series(16*x/(1-(1-2*x)^8), x, n+2), x, n), n = 0..30); # G. C. Greubel, Sep 17 2019

Mathematica

CoefficientList[Series[16*x/(1-(1-2*x)^8), {x,0,30}], x] (* G. C. Greubel, Sep 17 2019 *)

PARI

a(n)=polcoeff(16*x/(1-(1-2*x)^8)+x*O(x^n),n,x)

Sage

def A097194_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(16*x/(1-(1-2*x)^8)).list()
A097194_list(30) # G. C. Greubel, Sep 17 2019

Formula

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