OEIS/Infinite Products

OEIS collection of transforms

• Sequence transforms: Maple, Mathematica

weighout   b from a in 1+SUM b_n x^n = PI (1+x^n)^a_n
weighouti  a from b in 1+SUM b_n x^n = PI (1+x^n)^a_n
weighini   a from b in 1+SUM b_n x^n = PI (1+x^a_n)
weigh2out  b from a in 1+SUM b_n x^n = PI (x^-n+1+x^n)^a_n
weigh2outi a from b in 1+SUM b_n x^n = PI (x^-n+1+x^n)^a_n
weigh2in   b from a in 1+SUM b_n x^n = PI (x^-a_n+1+x^a_n)
weigh2ini  a from b in 1+SUM b_n x^n = PI (x^-a_n+1+x^a_n)

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).

List of common products

Maple:

seq(coeff(series(mul((1-3*x^k), k = 1..n), x, n+1), x, n), n=0..15);

• A000009 Product_{k>=1} (1+1*x^k)
• A032302 Product_{k>=1} (1+2*x^k)
• A032308 Product_{k>=1} (1+3*x^k)
• A261568 Product_{k>=1} (1+4*x^k)
• A261569 Product_{k>=1} (1+5*x^k)

• A010815 Product_{k>=1} (1-1*x^k)
• A000041 Product_{k>=1} (1-1*x^k)^(-1)
• A070877 Product_{k>=1} (1-2*x^k)
• A070933 Product_{k>=1} (1-2*x^k)^(-1)
• A292128 Product_{k>=1} (1-3*x^k)
• A242587 Product_{k>=1} (1-3*x^k)^(-1)
• A246936 Product_{k>=1} (1-4*x^k)^(-1)
• A246937 Product_{k>=1} (1-5*x^k)^(-1)

• A006906 Product_{k>=1} (1-1*k*x^k)^(-1)
• A265951 Product_{k>=1} (1-2*k*x^k)^(-1)
• A265974 Product_{k>=1} (1-3*k*x^k)^(-1)
• A265975 Product_{k>=1} (1-4*k*x^k)^(-1)
• A265976 Product_{k>=1} (1-5*k*x^k)^(-1)
• A022629 Product_{k>=1} (1+1*k*x^k)
• A022693 Product_{k>=1} (1+1*k*x^k)^(-1)
• A032309 Product_{k>=1} (1+2*k*x^k)