OEIS/Infinite Products

Programs

A046042 Number of partitions of n into fourth powers.

(Maple) 
g := -1 + 1/product(1 - x^(j^4), j=1..10): 
gser := series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102);
(Maple, A303350)
seq(coeff(series(mul((1+4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40);
(Mathematica) 
g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}]; 
gser = Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}]
(PARI, A322199)
{a(n) = polcoeff( 1/prod(m=1, n, 1 - (2^m+1)*x^m +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ","))

Generalized Euler transform

• defined by Seiichi Manyama in A266964

Product_{k>0} (1 - g(k)*x^k)^(- f(k)) = a(0) + a(1)*x + a(2)*x^2 + ...

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 (A001478) and g(n) = n (A000027), we get A266964.
4. With the default f(n) = g(n) = A000012 (all 1's), we get A000041 (number of partitions of n).

Interesting sequences

• A000081 Euler transform of itself, shifted by 1