OEIS/Infinite Products: Difference between revisions

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[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}]
=== Generalized Euler transform ===
=== Generalized Euler transform ===
* defined by 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]
  Product_{k>0} (1 - g(k)*x^k)^(- f(k)) = a(0) + a(1)*x + a(2)*x^2 + ...
  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 https://oeis.org/wiki/Euler_transform Euler transform].
# If we set g(n) = 1, we get the usual '''[https://oeis.org/wiki/Euler_transform Euler transform]'''.
# If we set f(n) = - h(n) and g(n) = -1, we get the weighout transform (cf. [https://oeis.org/A026007 A026007]).
# If we set f(n) = - h(n) and g(n) = -1, we get the weighout transform (cf. [https://oeis.org/A026007 A026007]).
# 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).

Revision as of 09:01, 5 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}]

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).