A145707 -id:A145707 - cp's OEIS frontend

A145704 Expansion of q^(1/4) * (eta(q^8) * eta(q^10) + eta(q^2) * eta(q^40)) / (eta(q^4) * eta(q^20)) in powers of q.

1, 1, 0, -1, 1, 0, 0, -1, 1, 1, -1, -1, 2, 1, -1, -1, 2, 2, -1, -2, 3, 3, -2, -3, 4, 3, -2, -4, 5, 4, -4, -5, 6, 6, -5, -6, 8, 7, -6, -8, 11, 10, -8, -11, 13, 11, -10, -13, 16, 15, -14, -17, 20, 18, -17, -20, 24, 23, -21, -25, 31, 29, -26, -32, 37, 34, -32

Offset: 0

Michael Somos, Oct 17 2008, Nov 11 2008, Jan 21 2009

sign

Denoted by "(160~b)" in Simon Norton's replicable function list.

			G.f. = 1 + x - x^3 + x^4 - x^7 + x^8 + x^9 - x^10 - x^11 + 2*x^12 + x^13 + ...
G.f. = 1/q + q^3 - q^11 + q^15 - q^27 + q^31 + q^35 - q^39 - q^43 + 2*q^47 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A145705, A145706, A145707.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^8] QPochhammer[ x^10] + x QPochhammer[ x^2] QPochhammer[ x^40]) / (QPochhammer[ x^4] QPochhammer[ x^20]), {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^10 + A) + x * eta(x^2 + A) * eta(x^40 + A)) / (eta(x^4 + A) * eta(x^20 + A)), n))};

G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).

a(n) = (-1)^n * A145705(n). a(2*n) = A145706(n). a(2*n + 1) = A145707(n).

A145703 Expansion of chi(x) / chi(-x^10) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 10, 11, 11, 13, 15, 17, 18, 20, 23, 25, 29, 32, 34, 39, 42, 47, 52, 56, 62, 68, 77, 83, 89, 99, 108, 119, 129, 139, 154, 167, 183, 199, 214, 234, 253, 276, 299, 322, 350, 378, 413, 445, 476, 518, 559

Offset: 0

Michael Somos, Oct 17 2008

nonn

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

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...
G.f. = q^3 + q^11 + q^27 + q^35 + q^43 + q^51 + q^59 + 2*q^67 + 2*q^75 + ...

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. A139632, A145702, A145707.

nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1)) / (1 - x^(20*k-10)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^10, x^10], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))};

Expansion of q^(-3/8) * eta(q^2)^2 * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10) ) in powers of q.
Euler transform of period 20 sequence [ 1, -1, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A145702.
G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(20*k - 10)).
a(n) = (-1)^n * A145707(n) = A139632(2*n + 1).
a(n) ~ exp(Pi*sqrt(n/5)) / (4*5^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, 0, -2, 0, 1, -1, 0, 0, 2, 0, 1, -1, 0, -1, 0, -3, 0, 1, -1, 0, -1, 2, -1, 4, 0, 1, -1, 0, -1, 1, -2, 1, -5, 0, 1, -1, 0, -1, 1, 0, 1, -1, 6, 0, 1, -1, 0, -1, 1, -1, 0, -2, 1, -8, 0, 1, -1, 0, -1, 1, -1, 2, -1, 4, 0, 10, 0, 1, -1, 0, -1, 1, -1, 1, -2, 1, -4, 0, -12, 0

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

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

Columns k=1-10 give: A000007, A081360, A109389, A261734, A133563, A261736, A113297, A261735, A261733, A145707.
Main diagonal gives A081362.
Cf. A286653, A286656, A290307.

Table[Function[k, SeriesCoefficient[Product[(1 + x^(k i))/(1 + x^i), {i, 1, n}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-1, x^k]/QPochhammer[-1, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

For asymptotics of column k see comment from Vaclav Kotesovec in A145707.

cp's OEIS Frontend

A145704 Expansion of q^(1/4) * (eta(q^8) * eta(q^10) + eta(q^2) * eta(q^40)) / (eta(q^4) * eta(q^20)) in powers of q.

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A145703 Expansion of chi(x) / chi(-x^10) in powers of x where chi() is a Ramanujan theta function.

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A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

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