OEIS/Infinite Products: Difference between revisions

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=== Infinite Products in your CAS ===
=== OEIS collection of transforms ===
* [https://oeis.org/wiki/Sequence_transforms Sequence transforms]: [https://oeis.org/transforms.txt Maple], [https://oeis.org/seqtranslib.txt 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 ===
[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.
[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.
  (Maple)  
  (Maple)  
  g := -1 + 1/product(1 - x^(j^4), j=1..10):  
  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);
  gser := series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102);
(Maple, [https://oeis.org/A303350 A303350])
seq(coeff(series(mul((1+4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40);
  (Mathematica)  
  (Mathematica)  
  g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}];  
  g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}];  
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  {a(n) = polcoeff( 1/prod(m=1, n, 1 - (2^m+1)*x^m +x*O(x^n)), n)}
  {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), ","))
  for(n=0, 30, print1(a(n), ","))
=== Generalized Euler transform ===
=== Generalized Euler transform ===
* defined by [https://oeis.org/wiki/User:Seiichi_Manyama Seiichi Manyama] in [https://oeis.org/A266964 A266964]
* defined by [https://oeis.org/wiki/User:Seiichi_Manyama Seiichi Manyama] in [https://oeis.org/A266964 A266964]
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# If we set f(n) = - n ([https://oeis.org/A001478 A001478]) and g(n) = n ([https://oeis.org/A000027 A000027]), we get A266964.
# If we set f(n) = - n ([https://oeis.org/A001478 A001478]) and g(n) = n ([https://oeis.org/A000027 A000027]), we get A266964.
# 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).
=== Interesting sequences ===
=== List of common products ===
* [https://oeis.org/A000081 A000081] Euler transform of itself, shifted by 1
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/A032302 A032302] Product_{k>=1} (1+2*x^k)
* [https://oeis.org/A032308 A032308] Product_{k>=1} (1+3*x^k)
* [https://oeis.org/A261568 A261568] Product_{k>=1} (1+4*x^k)
* [https://oeis.org/A261569 A261569] Product_{k>=1} (1+5*x^k)
 
* [https://oeis.org/A010815 A010815] Product_{k>=1} (1-1*x^k)
* [https://oeis.org/A000041 A000041] Product_{k>=1} (1-1*x^k)^(-1)
* [https://oeis.org/A070877 A070877] Product_{k>=1} (1-2*x^k)
* [https://oeis.org/A070933 A070933] Product_{k>=1} (1-2*x^k)^(-1)
* [https://oeis.org/A292128 A292128] Product_{k>=1} (1-3*x^k)
* [https://oeis.org/A242587 A242587] Product_{k>=1} (1-3*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)
====(1+m*k*x^k)====
* [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 A265974] Product_{k>=1} (1-3*k*x^k)^(-1)
* [https://oeis.org/A265951 A265975] Product_{k>=1} (1-4*k*x^k)^(-1)
* [https://oeis.org/A265951 A265976] Product_{k>=1} (1-5*k*x^k)^(-1)
* [https://oeis.org/A032309 A022629] Product_{k>=1} (1+1*k*x^k)
* [https://oeis.org/A032309 A022693] Product_{k>=1} (1+1*k*x^k)^(-1)
* [https://oeis.org/A032309 A032309] Product_{k>=1} (1+2*k*x^k)

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

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);

(1+m*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)
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