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%I #16 May 22 2022 04:12:34
%S 1,3,-3,9,-36,162,-783,3969,-20817,112023,-615033,3431403,-19398690,
%T 110880900,-639730305,3720657807,-21790419444,128398625658,
%U -760668489729,4528069760691,-27070491820644,162464919528222,-978463778897637,5911727071716891,-35821932198013809
%N 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.
%H G. C. Greubel, <a href="/A100239/b100239.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: A(x) = (1+3*x+sqrt(1+6*x-3*x^2))/2.
%F Given g.f. A(x), then B(x) = A(x) - (1+2*x) series reversion is -B(-x). - _Michael Somos_, Sep 07 2005
%F 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
%F a(n) = 3^n*[n<2] + 3*(-1)^(n+1)*A107264(n-2). - _G. C. Greubel_, May 21 2022
%e From the table of powers of A(x), we see that
%e 3^n+1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
%e A^1 = [1, 3], -3, 9, -36, 162, -783, 3969, -20817, 112023, ...
%e A^2 = [1, 6, 3], 0, -9, 54, -297, 1620, -8910, 49572, ...
%e A^3 = [1, 9, 18, 0], 0, 0, -27, 243, -1701, 10935, ...
%e A^4 = [1, 12, 42, 36, -9], 0, 0, 0, -81, 972, ...
%e A^5 = [1, 15, 75, 135, 45, -27], 0, 0, 0, 0, ...
%e A^6 = [1, 18, 117, 324, 324, 0, -54], 0, 0, 0, ...
%e A^7 = [1, 21, 168, 630, 1071, 567, -189, -81], 0, 0, ...
%e A^8 = [1, 24, 228, 1080, 2610, 2808, 540, -648, -81], 0, ...
%e the main diagonal of which is:
%e [x^n]A(x)^(n+1) = (n+1)*A057083(n) for n>=0.
%t 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}];
%t Table[a[n], {n,0,40}] (* _G. C. Greubel_, May 21 2022 *)
%o (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)
%o (PARI) a(n)=polcoeff((1+3*x+sqrt(1+6*x-3*x^2+x^2*O(x^n)))/2,n)
%o (SageMath)
%o def A100239_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+3*x+sqrt(1+6*x-3*x^2))/2 ).list()
%o A100239_list(40) # _G. C. Greubel_, May 21 2022
%Y Cf. A057083, A100226, A100239, A107264.
%K sign
%O 0,2
%A _Paul D. Hanna_, Nov 30 2004
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