A000727 -id:A000727 - cp's OEIS frontend

A022579 Expansion of Product_{m>=1} (1+x^m)^14.

1, 14, 105, 574, 2576, 10052, 35273, 113794, 342699, 974176, 2635955, 6833540, 17061345, 41197422, 96544003, 220212384, 490104727, 1066552228, 2273590095, 4755188704, 9771319068, 19751596934, 39317784863, 77150246040, 149357609184, 285497384004, 539227765104, 1006978117880

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=14 of A286335. Cf. A000707, A023003.

Coefficients(&*[(1+x^m)^14:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

nmax=50; CoefficientList[Series[Product[(1+q^m)^14,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^14)) \\ G. C. Greubel, Feb 25 2018

a(n) ~ (7/6)^(1/4) * exp(Pi * sqrt(14*n/3)) / (256 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (14/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f. A(x) = (1/2)*( G(sqrt(x)) + G(-sqrt(x)) )/G(x^4), where G(x) = Product_{n >= 1} 1/(1 - x^n)^4 is the g.f. of A023003 (see also A000727). - Peter Bala, Oct 05 2023

A280328 Expansion of f(-x)^3 * f(-x^2) * chi(-x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, -1, 5, 8, 1, -28, -11, 10, 41, 41, -26, -53, -84, 21, 101, 76, -3, -129, -99, 14, 190, 187, -59, -299, -263, 62, 336, 340, -27, -459, -370, 111, 645, 518, -228, -774, -806, 179, 973, 882, -147, -1233, -955, 291, 1565, 1325, -395, -1883, -1767, 338, 2318

Offset: 0

Michael Somos, Dec 31 2016

sign

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

			G.f. = 1 - 3*x - x^2 + 5*x^3 + 8*x^4 + x^5 - 28*x^6 - 11*x^7 + 10*x^8 + ...
G.f. = q^-1 - 3*q^5 - q^11 + 5*q^17 + 8*q^23 + q^29 - 28*q^35 - 11*q^41 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf.A000727, A280384.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 QPochhammer[ x^2] QPochhammer[ x^3, x^6]^3, {x, 0, n}];

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

Expansion of q * eta(q^6)^3 * eta(q^12) * eta(q^18)^3 / eta(q^36)^3 in powers of q^6.
Euler transform of period 6 sequence [-3, -4, -6, -4, -3, -4, ...].
a(n) = (-1)^n * A280384(n).
a(5*n + 1) / a(1) == A000727(n) (mod 5). a(125*n + 21) / a(21) == A000727(n) (mod 25).

A258779 Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -5, -10, 9, 14, -10, 0, 14, 2, -11, -32, 0, 14, -9, 26, 2, 0, 16, -22, 14, 0, 0, 26, -17, -32, -22, -10, -34, 14, 45, 38, 0, -34, 38, -22, 2, 0, -10, 64, -20, 0, 0, 0, -23, -46, 16, 0, -46, -32, 26, -10, 25, 18, 0, 38, 50, 0, 0, -22, -80, 50, 0, 26, 2

Offset: 0

Michael Somos, Jun 09 2015

sign

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

			G.f. = 1 + 2*x - 5*x^2 - 10*x^3 + 9*x^4 + 14*x^5 - 10*x^6 + 14*x^8 + ...
G.f. = q + 2*q^13 - 5*q^25 - 10*q^37 + 9*q^49 + 14*q^61 - 10*q^73 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000727, A106508, A143378, A187076, A187149.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] EllipticTheta[ 3, 0, x])^2, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A)^2))^2, n))};

Expansion of q^(-1/12) * (eta(q^2)^5 / (eta(q) * eta(q^4)^2))^2 in powers of q.
Euler transform of period 4 sequence [ 2, -8, 2, -4, ...].
a(n) = A000727(2*n) = A187076(2*n) = A106508(4*n) = A187149(4*n).
Convolution square of A143378.

A272202 Number of solutions of the congruence y^2 == x^3 - 1 (mod p) as p runs through the primes.

Original entry on oeis.org

2, 3, 5, 3, 11, 11, 17, 27, 23, 29, 27, 47, 41, 51, 47, 53, 59, 47, 51, 71, 83, 75, 83, 89, 83, 101, 123, 107, 107, 113, 147, 131, 137, 123, 149, 147, 143, 171, 167, 173, 179, 155, 191, 191, 197, 171, 195, 195, 227, 251, 233, 239, 227, 251, 257, 263, 269, 243, 251, 281

Offset: 1

Wolfdieter Lang, May 05 2016

In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the last column, starting with Conductor 144, as a strong Weil curve for the weight 2 newform  eta^{12}(12*z) / (eta^4(6*z) * eta^4(24*z)), symbolically 12^{12} 6^{-4} 24^{-4}, with Im(z) > 0, and the Dedekind eta function. See A187076 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z/3)* eta(2*z)^{12} / (eta^4(z)*eta^4(4*z)). For the q-expansion of 12^{12} 6^{-4} 24^{-4} one has a leading zero and 5 interspersed 0's: 0,1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,-8,...
The discriminant of this elliptic curve is -3^3 = -27.
For the elliptic curve y^2 == x^3 + 1 (mod prime(n)) see A000727, A272197, A272198, A272200 and A272201.

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2 == x^3 - 1 (mod prime(n)) begin:
n, prime(n), a(n)\ solutions (x, y)
1,    2,      2:   (0, 1), (1, 0)
2,    3,      3:   (1, 0), (2, 1), (2, 2)
3,    5,      5:   (0, 2), (0, 3), (1, 0),
                   (3, 1), (3, 4)
4,    7,      3:   (1, 0), (2, 0), (4, 0)
5,    11,    11:   (1, 0), (3, 2), (3, 9),
                   (5, 5), (5, 6), (7, 1),
                   (7, 10), (8, 4), (8, 7),
                   (10, 3), (10, 8)
...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A000040, A187076, A272203.

a(n) gives the number of solutions of the congruence y^2 == x^3 - 1 (mod prime(n)), n >= 1.

A173030 Expansion of q^(-1/6) * (eta(q)^4 + 7 * eta(q^7)^4) in powers of q.

Original entry on oeis.org

1, 3, 2, 8, -5, -4, -10, 8, -19, 0, 14, -16, -10, -4, 0, 6, 14, 20, 2, 0, -11, 20, 24, -16, 0, -4, 14, 8, -9, -15, 26, 0, 2, -28, 0, -16, -12, -28, -22, 0, 14, 16, 0, -30, 0, -28, 26, 32, -17, 0, 24, -16, -22, 0, -10, 32, -34, 55, 14, 0, 45, -4, 38, 8, 0, 0, -34, -8, 38, 0, -22, 42, 2, -28, 0, 0, -10, 20, -48, -40, -20, 44

Offset: 0

Michael Somos, Feb 07 2010

sign

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

			G.f. = 1 + 3*x + 2*x^2 + 8*x^3 - 5*x^4 - 4*x^5 - 10*x^6 + 8*x^7 - 19*x^8 + ...
G.f. = q + 3*q^7 + 2*q^13 + 8*q^19 - 5*q^25 - 4*q^31 - 10*q^37 + 8*q^43 + ...

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. A000727.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 + 7 x QPochhammer[ x^7]^4, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 + 7 * x * eta(x^7 + A)^4, n))};

Expansion of f(-x)^4 + 7 * x * f(-x^7)^4 = chi(-x) * chi(-x^7) * (psi(x)^4 + 7 * x^3 * psi(x^7)^4) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
Expansion of (phi(-x)^4 + 7 * phi(-x^7)^4) / (8 * chi(-x) * chi(-x^7)) in powers of x^2 where phi(), chi(), f() are Ramanujan theta functions.
a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (-p)^(e/2) (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = 252 (t/i)^2 * f(t) where q = exp(2 Pi i t).
A000727(n) = a(n) if n != 1 (mod 7). A000727(7*n + 1) + 7 * A000727(n) = a(7*n + 1).

A272199 Expansion of 1/(1 - 2x + 13x^2).

Original entry on oeis.org

1, 2, -9, -44, 29, 630, 883, -6424, -24327, 34858, 385967, 318780, -4380011, -12904162, 31131819, 230017744, 55321841, -2879586990, -6478357913, 24477915044, 133174482957, -51863929658, -1834996137757, -2995761189960, 17863427410921, 74671750291322

Offset: 0

Wolfdieter Lang, Apr 27 2016

a(n) gives the coefficient c(13^n) of (eta(z^6))^4, a modular cusp form of weight 2, when expanded in powers of q = exp(2*Pi*i*z), Im(z) > 0, assuming alpha-multiplicativity (not valid for p = 2 and 3) with alpha(x) = x (weight 2) and input c(13) = +2. Eta is the Dedekind function. See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 130, eq. (39) with k=2. See also A000727 where a(n)=c(13^n) = A000727((13^n-1)/6)=A000727(2*A091030(n)), n >= 0. For the proof that alpha-multiplicativity leads to the recurrence involving Chebyshev's S polynomials see a comment on A168175, and the Apostol reference, Exercise 6., p. 139.

			a(2) = c(13^2) = A000727(2*A091030(2)) =
A000727(28) = -9.

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 130, 138 - 139.

Indranil Ghosh, Table of n, a(n) for n = 0..1790
Index entries for linear recurrences with constant coefficients, signature (2,-13).
Index entries for sequences related to Chebyshev polynomials.

Cf. A023000, A049310, A168175.

I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1)-13*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 25 2016

CoefficientList[Series[1/(1 - 2 x + 13 x^2), {x, 0, 25}], x] (* Michael De Vlieger, Apr 27 2016 *)
LinearRecurrence[{2, -13}, {1, 2}, 30] (* Vincenzo Librandi, Nov 25 2016 *)

Vec(1/(1-2*x+13*x^2) + O(x^99)) \\ Altug Alkan, Apr 28 2016

G.f.: 1/(1 - 2*x + 13*x^2).
a(n) = 2*a(n-1) - 13*a(n-2), a(-1) = 0, a(0) = 1.
a(n) = sqrt(13)^n * S(n, 2/sqrt(13)), n >= 0, with Chebyshev's S polynomials (A049310).
a(n) = (Ap^(n+1) - Am^(n+1))/(Ap - Am) with Ap:= 1 + 2*sqrt(3)*i and Am = 1 - 2*sqrt(3)*i, (Binet - de Moivre formula), where i is the imaginary unit.
E.g.f.: (sqrt(3)*sin(2*sqrt(3)*x) + 6*cos(2*sqrt(3)*x))*exp(x)/6. - Ilya Gutkovskiy, Apr 27 2016

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A022579 Expansion of Product_{m>=1} (1+x^m)^14.

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A280328 Expansion of f(-x)^3 * f(-x^2) * chi(-x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.

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A258779 Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.

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A272202 Number of solutions of the congruence y^2 == x^3 - 1 (mod p) as p runs through the primes.

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A173030 Expansion of q^(-1/6) * (eta(q)^4 + 7 * eta(q^7)^4) in powers of q.

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A272199 Expansion of 1/(1 - 2*x + 13*x^2).

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A272199 Expansion of 1/(1 - 2x + 13x^2).