OEIS/Infinite Products: Difference between revisions
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# With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n). | # With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n). | ||
=== List of common products === | === List of common products === | ||
Maple: | Maple: <code>seq(coeff(series(mul((1-3*x^k), k = 1..n), x, n+1), x, n), n=0..15);</code> | ||
====(1+m*x^k)==== | |||
* [https://oeis.org/A000009 A000009] Product_{k>=1} (1+1*x^k) | * [https://oeis.org/A000009 A000009] Product_{k>=1} (1+1*x^k) | ||
* [https://oeis.org/A032302 A032302] Product_{k>=1} (1+2*x^k) | * [https://oeis.org/A032302 A032302] Product_{k>=1} (1+2*x^k) | ||
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* [https://oeis.org/A246936 A246936] Product_{k>=1} (1-4*x^k)^(-1) | * [https://oeis.org/A246936 A246936] Product_{k>=1} (1-4*x^k)^(-1) | ||
* [https://oeis.org/A246937 A246937] Product_{k>=1} (1-5*x^k)^(-1) | * [https://oeis.org/A246937 A246937] Product_{k>=1} (1-5*x^k)^(-1) | ||
====(1+m*k*x^k)==== | |||
* [https://oeis.org/A265951 A006906] Product_{k>=1} (1-1*k*x^k)^(-1) | * [https://oeis.org/A265951 A006906] Product_{k>=1} (1-1*k*x^k)^(-1) | ||
* [https://oeis.org/A265951 A265951] Product_{k>=1} (1-2*k*x^k)^(-1) | * [https://oeis.org/A265951 A265951] Product_{k>=1} (1-2*k*x^k)^(-1) | ||
Latest revision as of 13:31, 15 December 2023
OEIS collection of transforms
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 + ...
- If we set g(n) = 1, we get the usual Euler transform.
- If we set f(n) = - h(n) and g(n) = -1, we get the weighout transform (cf. A026007).
- If we set f(n) = - n (A001478) and g(n) = n (A000027), we get A266964.
- 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);
(1+m*x^k)
- 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)
(1+m*k*x^k)
- 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)