A375107 -id:A375107 - 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)).

A375150 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+5)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 01 2024

sign

Cf. A374900, A375106, A375148, A375149.

Cf. A375107.

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

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

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

G.f.: Sum_{k in Z} x^(5*k) / (1 - x^(7*k+1)).

A375108 Expansion of Sum_{k in Z} x^(3k) / (1 - x^(7k+3)).

Original entry on oeis.org

1, -1, 0, 2, 0, -2, 2, 0, 0, 0, 0, 0, 2, -1, 0, 2, -1, -2, 2, 0, 0, 0, 0, 2, 2, -2, 0, 0, 0, -2, 2, 0, 0, 2, 0, 0, 2, -2, -2, 2, 1, -2, 2, 2, 0, -1, 0, 0, 2, -4, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, -2, 0, 2, 0, -2, 2, 0, 0, 0, 0, -2, 2, 0, 2, 2, 0, -2, 2, 0, 0, -1, -2, 2, 2, -2, 0, 2, -1, -2, 2, 2, 0, 0, 0

Offset: 0

Seiichi Manyama, Jul 30 2024

sign

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. A375106, A375107.

my(N=100, x='x+O('x^N)); Vec(sum(k=-N, N, x^(3*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))/((1-x^(7*k-3))*(1-x^(7*k-4)))^2))

G.f.: Product_{k>0} (1-x^(7*k))^2 * (1-x^(7*k-1)) * (1-x^(7*k-6)) / ((1-x^(7*k-3)) * (1-x^(7*k-4)))^2.

A375159 Expansion of Sum_{k in Z} x^(4k) / (1 - x^(7k+2)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 01 2024

sign

Cf. A375107, A375148, A375158.

Cf. A375149.

my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^(4*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-1))*(1-x^(7*k-6)))^2/(1-x^k)))

G.f.: Product_{k>0} (1-x^(7*k))^3 * ((1-x^(7*k-1)) * (1-x^(7*k-6)))^2 / (1-x^k).

G.f.: Sum_{k in Z} x^(2*k) / (1 - x^(7*k+4)).

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

A108482 Expansion of a modular function for Gamma(7).

Original entry on oeis.org

1, 1, 1, 0, -1, -1, 0, 1, 2, 1, -1, -3, -3, 0, 3, 5, 3, -2, -6, -6, -1, 6, 9, 5, -4, -12, -11, -1, 12, 18, 10, -7, -21, -21, -3, 20, 30, 17, -13, -37, -35, -4, 36, 53, 30, -20, -62, -59, -8, 57, 85, 47, -35, -101, -95, -11, 94, 138, 78, -54, -159, -150, -19, 145, 213, 118, -85, -247, -231, -27, 225, 330, 183, -128, -375

Offset: 0

Michael Somos, Jun 04 2005

sign

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 157.

Cf. A375106, A375107.

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

Given g.f. A(x), then B(x)=x^-3*A(x^7) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^7 -v^7 +u*v^3 +u^9*v^6 +u^2*v^6 +3*u^5*v^8 -u^3*v^2 -u^3*v^9 -u^4*v^5 -u^5*v -5*u^6*v^4 -3*u^7*v^7 -2*u^8*v^3.
G.f.: Product_{k>0} (1-x^(7k-3))(1-x^(7k-4))/((1-x^(7k-1))(1-x^(7k-6))).
G.f.: B(x) / C(x), where B(x) is the g.f. of A375106 and C(x) is the g.f. of A375107. - Seiichi Manyama, Aug 03 2024

A375158 Expansion of Sum_{k in Z} x^(2k) / (1 - x^(7k+2)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 01 2024

sign

Cf. A375107, A375148, A375159.

Cf. A374900, A375108.

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

my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2*(1-x^(7*k-3))*(1-x^(7*k-4))/((1-x^(7*k-2))*(1-x^(7*k-5)))^2))

G.f.: Product_{k>0} (1-x^(7*k))^2 * (1-x^(7*k-3)) * (1-x^(7*k-4)) / ((1-x^(7*k-2)) * (1-x^(7*k-5)))^2.

cp's OEIS Frontend

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

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A375150 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+5)).

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A375108 Expansion of Sum_{k in Z} x^(3*k) / (1 - x^(7*k+3)).

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A375159 Expansion of Sum_{k in Z} x^(4*k) / (1 - x^(7*k+2)).

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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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A108482 Expansion of a modular function for Gamma(7).

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A375158 Expansion of Sum_{k in Z} x^(2*k) / (1 - x^(7*k+2)).

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A375108 Expansion of Sum_{k in Z} x^(3k) / (1 - x^(7k+3)).

A375159 Expansion of Sum_{k in Z} x^(4k) / (1 - x^(7k+2)).

A375158 Expansion of Sum_{k in Z} x^(2k) / (1 - x^(7k+2)).