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

0,2

Comments

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=0.

Links

G. C. Greubel, Table of n, a(n) for n = 0..59

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!.

Example

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! + ...

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 *)

Prog

(PARI) a(n)=polcoeff((1+x+x^2+x*O(x^n))^(2^n), n)
(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)
(Magma)
m:=40;
gf:= func< x | (&+[Log(1 +2^j*x +4^j*x^2)^j/Factorial(j): j in [0..m+1]]) >;
R<x>:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( gf(x) )); // G. C. Greubel, Jul 27 2023
(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.<x> = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A136518_list(m) # G. C. Greubel, Jul 27 2023

Crossrefs

Cf. A027907, A136519.

Sequence in context: A305854 A234296 A049505 * A168369 A363206 A317342

Adjacent sequences: A136515 A136516 A136517 * A136519 A136520 A136521

Keyword

nonn

Author

Paul D. Hanna, Jan 02 2008

Status

approved