A375150 -id:A375150 - cp's OEIS frontend

A375106 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+3)).

1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 0, 1, 1, 3, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 2, 1, 1, 2, 1, 1, 2, 0, 1, 1, 1, 0, 2, 2, 1, 0, 0, 1, 3, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 3, 1, 1, 2, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 0

Offset: 0

Seiichi Manyama, Jul 30 2024

nonn

R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 286.

Cf. A374900, A375148, A375149, A375150.

Cf. A375107, A375108.

my(N=100, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+3))))

my(N=100, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2/((1-x^(7*k-1))*(1-x^(7*k-6)))))

G.f.: Product_{k>0} (1-x^(7*k))^2 / ((1-x^(7*k-1)) * (1-x^(7*k-6))).

G.f.: Sum_{k in Z} x^(3*k) / (1 - x^(7*k+1)).

A375148 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+2)).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 0, 1, 1, 1, 3, 1, 2, 0, 2, 1, 2, 1, 1, 0, 2, 2, 2, 0, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 0, 2, 1, 2, 1, 4, 1, 2, 0, 0, 1, 3, 1, 0, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 0, 1, 0, 3, 2, 1, 1, 0, 2, 2, 1, 4, 0, 0

Offset: 0

Seiichi Manyama, Aug 01 2024

nonn

Cf. A374900, A375106, A375149, A375150.

Cf. A033687, A340453.

my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+2))))

my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^3*((1-x^(7*k-3))*(1-x^(7*k-4)))^2/(1-x^k)))

G.f.: Product_{k>0} (1-x^(7*k))^3 * ((1-x^(7*k-3)) * (1-x^(7*k-4)))^2 / (1-x^k).

G.f.: Sum_{k in Z} x^(2*k) / (1 - x^(7*k+1)).

A375149 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+4)).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 3, 1, -1, 1, 2, 0, 0, 1, 1, 1, 1, 1, 2, 1, -1, 1, 2, 0, 1, 0, 0, 1, 2, 0, 3, 1, -1, 1, 2, 0, 1, 1, 0, 2, 1, 0, 0, 1, -1, 1, 3, 1, 1, 1, 2, 0, 0, 0, 3, 1, -1, 1, 2, -1, 1, 2, 0, 1, 0, 0, 2, 1, 0, 1, 3, 0, 2, 1, 0, 1, 0, 0, 3, 1, -1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, -1, 0, 5

Offset: 0

Seiichi Manyama, Aug 01 2024

sign

Cf. A374900, A375106, A375148, A375150.

my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+4))))

my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^3*((1-x^(7*k-2))*(1-x^(7*k-5)))^2/(1-x^k)))

G.f.: Product_{k>0} (1-x^(7*k))^3 * ((1-x^(7*k-2)) * (1-x^(7*k-5)))^2 / (1-x^k).

G.f.: Sum_{k in Z} x^(4*k) / (1 - x^(7*k+1)).

A108483 Expansion of f(-x^2, -x^5) / f(-x, -x^6) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 0, 0, -1, 0, 1, 1, 0, -1, -1, -1, 0, 2, 2, 0, -1, -2, -2, 0, 3, 3, 0, -2, -3, -3, 0, 5, 5, 1, -3, -5, -5, 0, 7, 7, 1, -5, -8, -7, 1, 11, 12, 2, -7, -12, -11, 1, 15, 16, 3, -11, -18, -15, 2, 23, 24, 5, -15, -26, -22, 3, 31, 33, 7, -22, -37, -30, 5, 44, 47, 11, -30, -52, -42, 6, 59, 63, 15, -42, -72, -56, 10, 82, 88, 22

Offset: 0

Michael Somos, Jun 04 2005

sign

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

In Duke (2005) page 157 the g.f. is denoted by t(tau).

			G.f. = 1 + x - x^5 + x^7 + x^8 - x^10 - x^11 - x^12 + 2*x^14 + 2*x^15 + ...
G.f. = q^-2 + q^5 - q^33 + q^47 + q^54 - q^68 - q^75 - q^82 + 2*q^96 + ...

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162.

Michael Somos, Introduction to Ramanujan theta functions [This is just section 1 of the next item, but it is mentioned in over 1000 sequences. - _N. J. A. Sloane_, Nov 13 2019]
Michael Somos, A Multisection of q-Series
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A229894, A375106, A375150.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7] / (QPochhammer[ x, x^7] QPochhammer[ x^6, x^7]), {x, 0, n}]; (* Michael Somos, Oct 03 2013 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 0, 0, 1, -1, 0}[[Mod[k, 7, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 03 2015 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^[ 0, -1, 1, 0, 0, 1, -1][k%7 + 1]), n))};

Euler transform of period 7 sequence [ 1, -1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Oct 03 2013
Given g.f. A(x), then B(q) = q^-2*A(q^7) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^3 - u^6 + 3*u^4*v + u^7*v^3 + u^2*v^9 + u^8*v^6 - 3*u^2*v^2 - 2*u*v^6 - 5*u^3*v^5 - u^5*v^4 - u^9*v^2 - u^4*v^8 - u^6*v^7.
G.f.: Product_{k>0} (1 - x^(7*k - 2)) * (1 - x^(7*k - 5)) / ((1 - x^(7*k - 1)) * (1 - x^(7*k - 6))).
a(n) = A229894(7*n). - Michael Somos, Oct 03 2013
G.f.: B(x) / C(x), where B(x) is the g.f. of A375106 and C(x) is the g.f. of A375150. - Seiichi Manyama, Aug 03 2024

A230322 Expansion of f(-x^3, -x^4) / f(-x^2,-x^5) in powers of x where f(,) is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, 0, 1, -1, 0, 0, 0, 1, -1, 1, -1, 0, 0, -1, 2, -1, 2, -2, 0, 0, -1, 3, -3, 3, -3, 1, 0, -2, 5, -4, 5, -5, 1, 0, -3, 7, -7, 8, -7, 2, 0, -5, 11, -10, 12, -11, 3, 1, -7, 15, -16, 17, -15, 5, 1, -11, 22, -22, 25, -22, 7, 2, -15, 31, -33, 35, -30, 11, 2, -22

Offset: 0

Michael Somos, Oct 16 2013

sign

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

			G.f. = 1 + x^2 - x^3 + x^7 - x^8 + x^9 - x^10 - x^13 + 2*x^14 - x^15 + ...
G.f. = 1/q + q^13 - q^20 + q^48 - q^55 + q^62 - q^69 - q^90 + 2*q^97 - ...

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. A229894, A375107, A375150.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^7] QPochhammer[ x^4, x^7] / (QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7]), {x, 0, n}]

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^[ 0, 0, -1, 1, 1, -1, 0][k%7+1], 1 + x * O(x^n)), n))}

Euler transform of period 7 sequence [ 0, 1, -1, -1, 1, 0, 0, ...].
G.f.: Product_{k>0} (1 - x^(7*k - 3)) * (1 - x^(7*k - 4)) / ((1 - x^(7*k - 2)) * (1 - x^(7*k - 5))).
- a(n) = A229894(7*n + 1).
G.f.: B(x) / C(x), where B(x) is the g.f. of A375150 and C(x) is the g.f. of A375107. - Seiichi Manyama, Aug 03 2024

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A375106 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+3)).

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A375148 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+2)).

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A375149 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+4)).

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A108483 Expansion of f(-x^2, -x^5) / f(-x, -x^6) in powers of x where f() is a Ramanujan theta function.

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A230322 Expansion of f(-x^3, -x^4) / f(-x^2,-x^5) in powers of x where f(,) is Ramanujan's two-variable theta function.

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