A002171 -id:A002171 - cp's OEIS frontend

A278720 The p-defect p - N(p) of the congruence y^2 == x^3 + 4*x (mod p) for primes p, where N(p) is the number of solutions given by A276730.

0, 0, -2, 0, 0, 6, 2, 0, 0, -10, 0, -2, 10, 0, 0, 14, 0, -10, 0, 0, -6, 0, 0, 10, 18, -2, 0, 0, 6, -14, 0, 0, -22, 0, 14, 0, 22, 0, 0, -26, 0, -18, 0, -14, -2, 0, 0, 0, 0, 30, 26, 0, -30, 0, 2, 0, -26, 0, -18, 10, 0, -34, 0, 0, 26, 22, 0, 18, 0, -10, 34, 0, 0, 14, 0, 0, -34, 38, 2, -6, 0, 30, 0, 34, 0, 0, -14, 42, 38, 0, 0, 0, 0, 0, 0, 0, -10, -22, 0, -42

Offset: 1

Wolfdieter Lang, Dec 11 2016

This sequence gives also the p-defects for the congruences y^2 == x^3 - x (mod p), y^2 == x^3 - 11*x - 14 (mod p) and y^2 == x^3 - 11*x + 14 (mod p). See the Cremona link, Table 1, N = 32. - Wolfdieter Lang, Dec 22 2016
This elliptic curve y^2 = x^3 + 4*x appears as strong Weil curve for the weight 2 newform (eta(4*tau)*eta(8*tau))^2 of level N=32, with Dedekind's eta function. See the Martin-Ono link, Theorem 2, p. 3173, the row with Conductor 32. See also A002171 for the expansion of this newform in powers of q^4 (but with different offset). The also Nr. 49 of the Martin Table 1.
From this L-series of this elliptic curve one has:
a(n) = 0 if prime(n) == 2 or 3 (mod 4). (see the conjecture by Robert Israel, Sep 28 2016 in A276730).
If prime(n) == 1 (mod 4) = A002144(m) (for a unique m = m(n)) then prime(n) = A(m)^2 + B(m)^2 with the odd A(m) = A002972(m) and the even B(m) = 2*A002973(m). It turns out that 4*A002144(m) - a(m^2) = (2*B(m))^2 for m=m(n), and the sign s(m) of a(m) is + if A(m) + B(m) == 1 (mod 4) and - if A(m) + B(m) == 3 (mod 4). For the primes == 1 (mod 4) leading to sign + or - see A279392 or A279393, respectively. One has thus s(m) = (-1)^((A(m)-1)/2 + B(m)/2). See the Martin-Ono formula for a_{32}(p) in Theorem 3, p. 3175. This leads to the a(n) formula given below.

			a(1) = 0  because prime(1) = 2 == 2 (mod 4).
a(2) = 0 because prime(2) = 3 == 3 (mod 4).
a(3) = -2 because prime(3) = 5 = A002144(1) = A002972(1)^2 + (2*A002973(1))^2 = 1^2 + 2^2. Hence 2*A(1) = 2*A002972(1) = 2, and the sign s(1) = - because A(1) + B(1) = 1 + 2*1 = 3 == 3 (mod 4).
a(6) = +6 because prime(6) = 13 = A002144(2) = A(2)^2 + B(2)^2 = 3^2 + (2*1)^2. Hence 2*A(2) = 6 and the sign is + because A(2) + B(2) = 3 + 2 = 5 == 1 (mod 4).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
George E. Andrews, Number Theory, see S(1) expression p. 135.
J. E. Cremona, Algorithms for Modular Elliptic Curves.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A000040, A002144, A002171, A002972, A002973, A279392, A279393.

a(n) =  my(p=prime(n)); -sum(k=1, p-3, kronecker(k*(k+1)*(k+2), p)); \\ Michel Marcus, Nov 06 2020

a(n) = 0 if prime(n) == 2 or 3 (mod 4) (this is conjecture II from above).
a(n) = s(m)*2*A(m) if prime(n) = A002144(m), with A(m) = A002972(m) and the sign s(m) = (-1)^((A(m)-1)/2 + B(m)/2).
a(n) = - Sum_{k=1..p-3} ((k*(k+1)*(k+2))/p) where (x/y) is the Kronecker symbol. - Michel Marcus, Nov 06 2020

A319455 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^2.

Original entry on oeis.org

1, 2, 7, 14, 35, 66, 140, 252, 485, 840, 1512, 2534, 4347, 7084, 11705, 18622, 29862, 46522, 72779, 111310, 170534, 256586, 386101, 572488, 848050, 1240974, 1812979, 2621486, 3782669, 5410360, 7720237, 10932740, 15443120, 21669546, 30327570, 42196022, 58555543, 80832850

Offset: 0

Ilya Gutkovskiy, Sep 19 2018

nonn

Convolution inverse of A002171.
Self-convolution of A002513.
Convolution of A000041 and A029862.
Euler transform of period 2 sequence [2, 4, ...].

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000041, A001934, A001936, A002171, A002513, A029862.

a:=series(mul(1/((1-x^k)*(1-x^(2*k)))^2,k=1..55),x=0,38): seq(coeff(a,x,n),n=0..37); # Paolo P. Lava, Apr 02 2019

nmax = 37; CoefficientList[Series[Product[1/((1 - x^k)*(1 - x^(2*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^2])^2, {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[2 Sum[(4 DivisorSigma[1, k] - DivisorSigma[1, 2 k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

seq(n)={Vec(exp(2*sum(k=1, n, (4*sigma(k) - sigma(2*k))*x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Sep 19 2018

G.f.: Product_{k>=1} (1 + x^k)^2/(1 - x^(2*k))^4.
G.f.: exp(2*Sum_{k>=1} (4*sigma(k) - sigma(2*k))*x^k/k).
a(n) ~ exp(Pi*sqrt(2*n)) / (2^(13/4)*n^(7/4)). - Vaclav Kotesovec, Sep 14 2021

A228072 Expansion of psi(x^2)^2 * phi(-x^2)^6 + 8 * x * psi(x^2)^6 * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 8, -10, 16, 37, -40, -50, -80, -30, 40, 128, 48, -25, 80, -34, 320, -320, -160, 310, -400, 410, 152, -370, -416, -87, -240, -410, 400, 320, -200, 30, 592, 500, 776, 384, 400, -630, -200, -640, -1120, -359, 552, 300, -272, -326, -800, 2560, -400, -110

Offset: 0

Michael Somos, Sep 02 2013

sign

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

			G.f. = 1 + 8*x - 10*x^2 + 16*x^3 + 37*x^4 - 40*x^5 - 50*x^6 - 80*x^7 - 30*x^8 + ...
G.f. = q + 8*q^3 - 10*q^5 + 16*q^7 + 37*q^9 - 40*q^11 - 50*q^13 - 80*q^15 - 30*q^17 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Hossein Movasati, Younes Nikdelan, Product formulas for weight two newforms, arXiv:1803.01414 [math.NT], 2018.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002171, A227239, A227317, A227695.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^12 + 8 x QPochhammer[ x^4]^12) / (QPochhammer[ x^2] QPochhammer[ x^4])^2, {x, 0, n}];

{a(n) = local(A); if( n<0, 0, A = x * O(x^n);polcoeff( (eta(x^2 + A)^5 / eta(x^4 + A))^2 + 8 * x * (eta(x^4 + A)^5 / eta(x^2 + A))^2, n))};

Expansion of q^(-1/2) * ((eta(q^2)^5 / eta(q^4))^2 + 8 * (eta(q^4)^5 / eta(q^2))^2) in powers of q.
Expansion of q^(-1/2) * (eta(q^2)^12 + 8 * eta(q^4)^12) / ( eta(q^2) * eta(q^4) )^2 in powers of q.
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^2 (t / i)^4 f(t) where q = exp(2 Pi i t).
a(2*n) = A227695(n). a(2*n + 1) = 8 * A227317(n).
If F(x) is the g.f. for A002171, then A(x) * F(x^2) = B(x) the g.f. for A227239. - Michael Somos, Jan 08 2015

A258739 Expansion of (f(-x)^3 / f(-x^2))^6 - 64 * x * (f(-x^2)^3 / f(-x))^6 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -82, -243, -1194, 2242, 0, 3599, 2950, 0, -12242, -20950, 19926, -16807, 7294, 0, 18950, 97908, 0, -88806, 0, 59049, -183844, 51050, 0, -92142, -98002, 0, 246486, 118706, 290142, -161051, -38868, 0, 0, 75658, 0, -241900, 47614, -544806, -493658, 0, 0

Offset: 0

Michael Somos, Jun 08 2015

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.
Denoted by g_6(q) in Cynk and Hulek on page 8 as a level 32 cusp form of weight 6.

			G.f. = 1 - 82*x - 243*x^2 - 1194*x^3 + 2242*x^4 + 3599*x^6 + 2950*x^7 + ...
G.f. = q - 82*q^5 - 243*q^9 - 1194*q^13 + 2242*q^17 + 3599*q^25 + 2950*q^29 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds (arXiv:math/0509424).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000729, A002171, A008441, A215472, A215601.

A := Basis( CuspForms( Gamma0(32), 6), 165); A[1]  - 82*A[5] - 243*A[9] - 1194*A[13] + 2242*A[16];

a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^3 / QPochhammer[ x^2])^6 - 64 x (QPochhammer[ x^2]^3 / QPochhammer[ x])^6, {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))^6 - 64 * x * (eta(x^2 + A)^3 / eta(x + A))^6, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if(n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if(p==2, 0, p%4==3, if(e%2, 0, (-p)^(5*e/2)), y = -sum(i=0, p-1, kronecker(i^3-i, p)); a0=2; a1=y; for(i=2, 5, x=y*a1 -p*a0; a0=a1; a1=x); y=a1; a0=1; a1=y; for(i=2, e, x=y*a1 -p^5*a0; a0=a1; a1=x); a1)))};

Expansion of q^(-1/4) * ((eta(-q)^3 / eta(-q^2))^6 - 64 * (eta(-q^2) / eta(-q))^6) in powers of q.
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(5*e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = -(32^3) (t/i)^6 f(t) where q = exp(2 Pi i t).

A261403 Coefficients of an example of a modular form of weight 2 for the group Gamma_0(32).

Original entry on oeis.org

1, 12, 4, 0, 0, -24, -16, 0, -8, -36, 24, 0, 0, 72, -32, 0, 24, 24, 52, 0, 0, 0, -48, 0, -32, -12, 56, 0, 0, -120, -96, 0, 24, 0, 72, 0, 0, -24, -80, 0, -48, 120, 128, 0, 0, 72, -96, 0, 96, -84, 124, 0, 0, 168, -160, 0, -64, 0, 120, 0, 0, -120, -128, 0, 24, -144, 192, 0, 0, 0, -192

Offset: 0

N. J. A. Sloane, Aug 20 2015

sign

This is a particular member of an eight-dimensional vector space.

Robin Visser, Table of n, a(n) for n = 0..1000
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472, 2015, See Eq. (B.88).

Cf. A002171.

def a(n):
    B = ModularForms(Gamma0(32),2).basis()
    f = B[1] + 12*B[0] + 4*B[3] - 16*B[6] - 8*B[7]
    return f.coefficient(n)  # Robin Visser, Dec 12 2023

More terms from Robin Visser, Dec 12 2023

A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)(1 - x^(2k)))^n.

Original entry on oeis.org

1, -1, -3, 14, -11, -81, 282, -57, -2043, 5405, 2417, -46476, 94522, 110512, -943407, 1505289, 2807589, -16888311, 23645199, 46006542, -265972791, 472882620, 187884672, -3981273597, 14234579226, -19187383356, -78662039004, 502118911904, -847583768679, -2627514175002

Offset: 0

Ilya Gutkovskiy, Sep 19 2018

sign

Cf. A002171, A002288, A008705, A030204, A030211, A066535, A296043, A296044, A319457.

Table[SeriesCoefficient[Product[((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - 4 DivisorSigma[1, k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]

a(n) = [x^n] Product_{k>=1} (1 - x^(2*k))^(2*n)/(1 + x^k)^n.

a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k).

cp's OEIS Frontend

A278720 The p-defect p - N(p) of the congruence y^2 == x^3 + 4*x (mod p) for primes p, where N(p) is the number of solutions given by A276730.

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A319455 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(2*k)))^2.

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A228072 Expansion of psi(x^2)^2 * phi(-x^2)^6 + 8 * x * psi(x^2)^6 * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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A258739 Expansion of (f(-x)^3 / f(-x^2))^6 - 64 * x * (f(-x^2)^3 / f(-x))^6 in powers of x where f() is a Ramanujan theta function.

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A261403 Coefficients of an example of a modular form of weight 2 for the group Gamma_0(32).

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A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)*(1 - x^(2*k)))^n.

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A319455 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^2.

A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)(1 - x^(2k)))^n.