A328880 -id:A328880 - cp's OEIS frontend

A328796 Expansion of chi(x) / chi(-x^6) in powers of x where chi() is a Ramanujan theta function.

1, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 8, 8, 11, 12, 12, 16, 17, 18, 23, 25, 26, 32, 35, 37, 45, 49, 52, 62, 67, 72, 85, 92, 98, 114, 124, 133, 153, 166, 178, 203, 220, 236, 268, 290, 311, 350, 379, 407, 456, 493, 529, 589, 636, 683, 758, 818, 877

Offset: 0

Michael Somos, Oct 27 2019

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square is A328790.
G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A328880.

			G.f. = 1 + x + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...
G.f. = q^5 + q^29 + q^77 + q^101 + q^125 + 2*q^149 + 2*q^173 + ...

Cristina Ballantine and Mircea Merca, 6-regular partitions: new combinatorial properties, congruences, and linear inequalities, arXiv:2302.01253 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A261736, A328790, A328800.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^6], {x, 0, n}];

{a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^12 + A)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};

Expansion of q^(-5/24) * (eta(q^2)^2 * eta(q^12)) / (eta(q) * eta(q^4) * eta(q^6)) in power of q.
Euler transform of period 12 sequence [1, -1, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, ...].
G.f.: Product_{k>=1} (1 + x^(6*k))/(1 + (-x)^k) = Product_{k>=1} (1 + x^(2*k-1)) * (1 + x^(6*k)).
A261736(n) = (-1)^n * a(n).
a(n) ~ exp(sqrt(2*n)*Pi/3) / (2^(7/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019

A333267 If n = Product (p_j^k_j) then a(n) = Product (a(pi(p_j)) * k_j), where pi = A000720.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 2, 1, 4, 2, 2, 3, 2, 2, 1, 2, 3, 2, 1, 3, 4, 1, 1, 1, 5, 1, 2, 2, 4, 2, 3, 1, 3, 1, 2, 2, 2, 2, 2, 1, 4, 4, 2, 2, 2, 4, 3, 1, 6, 3, 1, 2, 2, 2, 1, 4, 6, 1, 1, 3, 4, 2, 2, 2, 6, 2, 2, 2, 6, 2, 1, 1, 4, 4, 1, 2, 4, 2, 2, 1, 3, 3, 2, 2, 4, 1, 1, 3, 5, 2, 4, 2, 4

Offset: 1

Ilya Gutkovskiy, Mar 13 2020

			a(36) = a(2^2 * 3^2) = a(prime(1)^2 * prime(2)^2) = a(1) * 2 * a(2) * 2 = 4.

Cf. A000026, A000720, A002110, A003963, A005361, A054725, A109129, A276625 (positions of 1's), A282446, A304117, A318046, A328880.

a:= proc(n) option remember;
      mul(a(numtheory[pi](i[1]))*i[2], i=ifactors(n)[2])
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 13 2020

a[1] = 1; a[n_] := a[n] = Times @@ (a[PrimePi[#[[1]]]] #[[2]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 100}]

a(n) = A005361(n) * Product_{p|n, p prime} a(pi(p)).
a(n) = a(prime(n)).
a(p^k) = k * a(p), where p is prime.
a(A002110(n)) = Product_{k=1..n} a(k).

A341041 If n = Product (p_j^k_j) then a(n) = 1 + Product (a(pi(p_j))), where pi = A000720, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 2, 3, 4, 5, 3, 4, 3, 7, 2, 4, 3, 3, 4, 5, 5, 4, 3, 4, 4, 3, 3, 5, 7, 6, 2, 9, 4, 7, 3, 4, 3, 7, 4, 5, 5, 4, 5, 7, 4, 8, 3, 3, 4, 7, 4, 3, 3, 13, 3, 5, 5, 5, 7, 4, 6, 5, 2, 10, 9, 4, 4, 7, 7, 5, 3, 6, 4, 7, 3, 9, 7, 6, 4, 3, 5, 5, 5, 10, 4, 9, 5, 4, 7, 7, 4, 11, 8, 7, 3, 5, 3, 9, 4

Offset: 1

Ilya Gutkovskiy, Feb 03 2021

nonn

			a(45) = a(3^2 * 5) = a(prime(2)^2 * prime(3)) = 1 + a(2) * a(3) = 1 + 2 * 3 = 7.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A093320, A156061, A214567, A328880.

a:= n-> `if`(n=1, 1, 1+mul(a(numtheory[pi](i[1])), i=ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 03 2021

a[1] = 1; a[n_] := a[n] = 1 + Times @@ (a[PrimePi[#[[1]]]] & /@ FactorInteger[n]); Table[a[n], {n, 100}]

a(n) = a(prime(n)^k) - 1 for k > 0.

a(2*n) = a(n) for n > 1.

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A328796 Expansion of chi(x) / chi(-x^6) in powers of x where chi() is a Ramanujan theta function.

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A333267 If n = Product (p_j^k_j) then a(n) = Product (a(pi(p_j)) * k_j), where pi = A000720.

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A341041 If n = Product (p_j^k_j) then a(n) = 1 + Product (a(pi(p_j))), where pi = A000720, with a(1) = 1.

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