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

0,2

Links

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

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!.

Example

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.

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 *)

Prog

(PARI) a(n)=polcoeff((1+x+x^2+x*O(x^n))^(2^n+1), n)
(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)
(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<x>:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( gf(x) )); // G. C. Greubel, Jul 27 2023
(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.<x> = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A136519_list(m) # G. C. Greubel, Jul 27 2023

Crossrefs

Cf. A027907, A136518.

Sequence in context: A228901 A195226 A264558 * A102556 A016065 A005016

Adjacent sequences: A136516 A136517 A136518 * A136520 A136521 A136522

Keyword

nonn

Author

Paul D. Hanna, Jan 02 2008

Status

approved