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.
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
Offset: 0
Keywords
Examples
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, 112023, ... 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, ... A^5 = [1, 15, 75, 135, 45, -27], 0, 0, 0, 0, ... A^6 = [1, 18, 117, 324, 324, 0, -54], 0, 0, 0, ... A^7 = [1, 21, 168, 630, 1071, 567, -189, -81], 0, 0, ... A^8 = [1, 24, 228, 1080, 2610, 2808, 540, -648, -81], 0, ... the main diagonal of which is: [x^n]A(x)^(n+1) = (n+1)*A057083(n) for n>=0.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
a[n_]:= a[n]= 3^n*Boole[n<2] + 3*(-1)^(n+1)*Sum[Binomial[k+1, n-k-1]*Binomial[n-2,k]*3^k/(k+1), {k,0,n-2}]; Table[a[n], {n,0,40}] (* G. C. Greubel, May 21 2022 *) -
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)
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PARI
a(n)=polcoeff((1+3*x+sqrt(1+6*x-3*x^2+x^2*O(x^n)))/2,n)
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SageMath
def A100239_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P( (1+3*x+sqrt(1+6*x-3*x^2))/2 ).list() A100239_list(40) # G. C. Greubel, May 21 2022
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+2*x) 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
a(n) = 3^n*[n<2] + 3*(-1)^(n+1)*A107264(n-2). - G. C. Greubel, May 21 2022