A136518 a(n) = A027907(2^n, n), where A027907 = triangle of trinomial coefficients.
1, 2, 10, 112, 3620, 360096, 116950848, 129755798400, 507413158135840, 7132358041777380352, 364730093112968976177664, 68393665694364347188157159424, 47308574208170527265149009962117120
Offset: 0
Keywords
Examples
A(x) = 1 + 2*x + 10*x^2 + 112*x^3 + 3620*x^4 + 360096*x^5 + ... A(x) = 1 + log(1 +2*x +4*x^2) + log(1 +4*x +16*x^2)^2/2! + log(1 +8*x +64*x^2)^3/3! + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..59
Programs
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Magma
m:=40; gf:= func< x | (&+[Log(1 +2^j*x +4^j*x^2)^j/Factorial(j): j in [0..m+1]]) >; R
:=PowerSeriesRing(Rationals(), m); Coefficients(R!( gf(x) )); // G. C. Greubel, Jul 27 2023 -
Mathematica
With[{m=40, f= 1 +2^j*x +4^j*x^2}, CoefficientList[Series[ Sum[Log[f]^j/j!, {j,0,m+1}], {x,0,m}], x]] (* G. C. Greubel, Jul 27 2023 *) -
PARI
a(n)=polcoeff((1+x+x^2+x*O(x^n))^(2^n),n)
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PARI
/* As coefficient x^n of Series: */ a(n)=polcoeff(sum(i=0,n,log(1+2^i*x+2^(2*i)*x^2 +x*O(x^n))^i/i!),n)
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SageMath
m=40 def f(x): return sum( log(1 + 2^j*x + 4^j*x^2)^j/factorial(j) for j in range(m+2) ) def A136518_list(prec): P.
= PowerSeriesRing(QQ, prec) return P( f(x) ).list() A136518_list(m) # G. C. Greubel, Jul 27 2023
Formula
a(n) = [x^n] (1 + x + x^2)^(2^n), the coefficient of x^n in (1 + x + x^2)^(2^n).
O.g.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x + 4^n*x^2)^n / n!.
Comments