A136519 a(n) = A027907(2^n+1, n), where A027907 = triangle of trinomial coefficients.
1, 3, 15, 156, 4556, 417384, 128004240, 136874853504, 523288667468832, 7257782720507161152, 368292386875012729754240, 68761030015590030510485191680, 47447175348985315294381264871833600
Offset: 0
Keywords
Examples
A(x) = 1 + 3*x + 15*x^2 + 156*x^3 + 4556*x^4 + 417384*x^5 + ...
A(x) = (1 +x +x^2) + (1 +2*x +4*x^2)*log(1 +2*x +4*x^2) + (1 +4*x +16*x^2)*log(1 +4*x +16*x^2)^2/2! + (1 +8*x +64*x^2)*log(1 +8*x +64*x^2)^3/3! + (1 +16*x +256*x^2)*log(1 +16*x +256*x^2)^4/4! + ...
This is a special case of the more general statement:
Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =
Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b)
where F(x) = 1+x+x^2, q=2, m=1, b=1.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..59
Programs
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Magma
m:=40; // gf of A136519 gf:= func< x | (&+[(1 +2^j*x +4^j*x^2)*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[f*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+1),n)
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PARI
/* As coefficient x^n of Series: */ a(n)=polcoeff(sum(i=0,n,(1+2^i*x+2^(2*i)*x^2)*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( (1 + 2^j*x + 4^j*x^2)*log(1 + 2^j*x + 4^j*x^2)^j/factorial(j) for j in range(m+2) ) def A136519_list(prec): P.
= PowerSeriesRing(QQ, prec) return P( f(x) ).list() A136519_list(m) # G. C. Greubel, Jul 27 2023
Formula
a(n) = [x^n] (1 + x + x^2)^(2^n+1), the coefficient of x^n in (1 + x + x^2)^(2^n+1).
O.g.f.: A(x) = Sum_{n>=0} (1 + 2^n*x + 4^n*x^2) * log(1 + 2^n*x + 4^n*x^2)^n / n!.