OEIS/Infinite Products: Difference between revisions

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=== Infinite Products in your CAS ===
=== Infinite Products in your CAS ===
[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);
* Mathematica  
(Mathematica)
  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}]; Table[Coefficient[gser, x, n], {n, 1, 102}]
  gser = Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}]
(PARI, [https://oeis.org/A322199 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 ===
=== 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 ===
* [https://oeis.org/A000081 A000081] Euler transform of itself, shifted by 1

Revision as of 21:00, 7 December 2020

Infinite Products in your CAS

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);
(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).

Interesting sequences

  • A000081 Euler transform of itself, shifted by 1