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
- 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).