OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
FORMULA
a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} (Sum_{d|k} d^(2+k/d)) * a(n-k) for n > 0. - Seiichi Manyama, Nov 02 2017
From Seiichi Manyama, Nov 14 2017: (Start)
A generalized Euler transform.
Suppose given two sequences f(n) and g(n), n>0, we define a new sequence a(n), n>=0, by Product_{n>0} (1 - g(n)*x^n)^(-f(n)) = a(0) + a(1)*x + a(2)*x^2 + ...
Since Product_{n>0} (1 - g(n)*x^n)^(-f(n)) = exp(Sum_{n>0} (Sum_{d|n} d*f(d)*g(d)^(n/d))*x^n/n), we see that a(n) is given explicitly by a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d*f(d)*g(d)^(n/d).
Examples:
1. If we set g(n) = 1, we get the usual Euler transform.
2. If we set f(n) = -h(n) and g(n) = -1, we get the weighout transform (cf. A026007).
3. If we set f(n) = -n and g(n) = n, we get this sequence.
(End)
MAPLE
seq(coeff(series(mul((1-k*x^k)^k, k=1..n), x, n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 31 2018
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[(1-k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
(* More efficient program: *) nmax = 40; s = 1-x; Do[s*=Sum[Binomial[k, j]*(-1)^j*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; Take[CoefficientList[s, x], nmax]
PROG
(PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k*x^k)^k)) \\ Seiichi Manyama, Nov 18 2017
(Ruby)
def s(f_ary, g_ary, n)
s = 0
(1..n).each{|i| s += i * f_ary[i] * g_ary[i] ** (n / i) if n % i == 0}
s
end
def A(f_ary, g_ary, n)
ary = [1]
a = [0] + (1..n).map{|i| s(f_ary, g_ary, i)}
(1..n).each{|i| ary << (1..i).inject(0){|s, j| s + a[j] * ary[-j]} / i}
ary
end
def A266964(n)
A((0..n).map{|i| -i}, (0..n).to_a, n)
end
p A266964(50) # Seiichi Manyama, Nov 18 2017
(Magma) m:=50; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-k*q^k)^k: k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Vaclav Kotesovec, Jan 07 2016
STATUS
approved