A122792 -id:A122792 - cp's OEIS frontend

A293306 Expansion of (eta(q)*eta(q^3))/eta(q^2)^2 in powers of q.

1, -1, 1, -3, 4, -5, 6, -9, 13, -16, 20, -27, 36, -44, 54, -69, 88, -107, 130, -162, 200, -240, 288, -351, 426, -507, 602, -723, 864, -1019, 1200, -1422, 1681, -1968, 2300, -2700, 3160, -3674, 4266, -4965, 5768, -6665, 7692, -8892, 10260, -11792, 13536, -15552

Offset: 0

Seiichi Manyama, Oct 05 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See p. 15.

Main diagonal of A293305.

Cf. A122792.

nmax = 50; CoefficientList[Series[Product[(1 - x^k) * (1 - x^(3*k)) / (1 - x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2017 *)

G.f.: Product_{i>0} (1 + Sum_{j>0} (-1)^j*j*q^(j*i)).

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)). - Vaclav Kotesovec, Oct 05 2017

A293307 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^jjx^(j*i)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 0, 3, 0, 1, 1, 0, -1, 5, 0, 1, 1, 0, 2, -1, 7, 0, 1, 1, 0, 2, 5, 3, 11, 0, 1, 1, 0, 2, 1, 3, 3, 15, 0, 1, 1, 0, 2, 1, -5, 0, -1, 22, 0, 1, 1, 0, 2, 1, 0, 0, 11, -8, 30, 0, 1, 1, 0, 2, 1, 0, 10, 7, 25, -8, 42, 0, 1, 1, 0, 2, 1, 0

Offset: 0

Seiichi Manyama, Oct 05 2017

			Square array begins:
   1, 1,  1, 1,  1, ...
   0, 1,  1, 1,  1, ...
   0, 2,  0, 0,  0, ...
   0, 3, -1, 2,  2, ...
   0, 5, -1, 5,  1, ...
   0, 7,  3, 3, -5, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000007, A000041, A293294.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A122792.
Cf. A293293, A293305.

A139136 Expansion of psi(-q) / f(q^3) where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 0, -2, 1, 0, 4, -2, 0, -6, 4, 0, 10, -6, 0, -16, 9, 0, 24, -14, 0, -36, 20, 0, 52, -29, 0, -74, 42, 0, 104, -58, 0, -144, 80, 0, 198, -110, 0, -268, 148, 0, 360, -198, 0, -480, 264, 0, 634, -347, 0, -832, 454, 0, 1084, -592, 0, -1404, 764, 0, 1808, -982

Offset: 0

Michael Somos, Apr 10 2008

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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - q - 2*q^3 + q^4 + 4*q^6 - 2*q^7 - 6*q^9 + 4*q^10 + 10*q^12 - 6*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A122792, A132002, A139135.

a[ n_] := SeriesCoefficient[ 2^(-1/2) q^(-1/8) EllipticTheta[ 2, Pi/4, q^(1/2)] / QPochhammer[ -q^3], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A) * eta(x^6 + A)^3), n))};

Expansion of eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6)^3) in powers of q.
Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, 2, -1, -1, -2, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139135.
G.f.: Product_{k>0} P(12, x^k) / ( (1 + x^(2*k-1))^2 * P(3, x^k) * P(6, x^k)^2) where P(n, x) is n-th cyclotomic polynomial.
a(3*n) = A132002(n). a(3*n + 1) = - A139135(n). a(3*n + 2) = 0.
a(n) = (-1)^n * A122792(n). - Michael Somos, Sep 07 2015

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A293306 Expansion of (eta(q)*eta(q^3))/eta(q^2)^2 in powers of q.

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A293307 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^jjx^(j*i)).

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A139136 Expansion of psi(-q) / f(q^3) where psi(), f() are Ramanujan theta functions.

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cp's OEIS Frontend

A293306 Expansion of (eta(q)*eta(q^3))/eta(q^2)^2 in powers of q.

Views

Author

Keywords

Links

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Programs

Mathematica

Formula

A293307 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^j*j*x^(j*i)).

Views

Author

Keywords

Examples

Links

Crossrefs

A139136 Expansion of psi(-q) / f(q^3) where psi(), f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A293307 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^jjx^(j*i)).