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A100239 G.f. A(x) satisfies: 3^n + 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (3+z)^n + (1+z)^n - z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n. 3
1, 3, -3, 9, -36, 162, -783, 3969, -20817, 112023, -615033, 3431403, -19398690, 110880900, -639730305, 3720657807, -21790419444, 128398625658, -760668489729, 4528069760691, -27070491820644, 162464919528222, -978463778897637, 5911727071716891, -35821932198013809 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..24.

FORMULA

G.f.: A(x) = (1+3*x+sqrt(1+6*x-3*x^2))/2.

Given g.f. A(x), then B(x)=A(x)-1-2x series reversion is -B(-x). - Michael Somos, Sep 07 2005

Given g.f. A(x) and C(x) = g.f. of A025226, then B(x)=A(x)-1-2x satisfies B(x)=x-C(x*B(x)). - Michael Somos, Sep 07 2005

EXAMPLE

From the table of powers of A(x), we see that

3^n+1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:

A^1=[1,3],-3,9,-36,162,-783,3969,-20817,...

A^2=[1,6,3],0,-9,54,-297,1620,-8910,49572,...

A^3=[1,9,18,0],0,0,-27,243,-1701,10935,...

A^4=[1,12,42,36,-9],0,0,0,-81,972,-8262,...

A^5=[1,15,75,135,45,-27],0,0,0,0,-243,...

A^6=[1,18,117,324,324,0,-54],0,0,0,0,0,...

A^7=[1,21,168,630,1071,567,-189,-81],0,0,0,...

A^8=[1,24,228,1080,2610,2808,540,-648,-81],0,0,...

the main diagonal of which is:

[x^n]A(x)^(n+1) = (n+1)*A057083(n) for n>=0.

PROG

(PARI) a(n)=if(n==0, 1, (3^n+1-sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n+x*O(x^k), k)))/n)

(PARI) a(n)=polcoeff((1+3*x+sqrt(1+6*x-3*x^2+x^2*O(x^n)))/2, n)

CROSSREFS

Cf. A100226, A100239, A057083.

Sequence in context: A257620 A032086 A241278 * A245023 A038080 A257621

Adjacent sequences:  A100236 A100237 A100238 * A100240 A100241 A100242

KEYWORD

sign

AUTHOR

Paul D. Hanna, Nov 30 2004

STATUS

approved

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Last modified April 3 20:29 EDT 2020. Contains 333199 sequences. (Running on oeis4.)