A000727 -id:A000727 - cp's OEIS frontend

A258404 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^4 dx.

1, 6, 1, 8, 2, 0, 2, 4, 2, 2, 9, 4, 8, 5, 6, 5, 6, 1, 8, 0, 2, 6, 1, 3, 3, 4, 9, 8, 5, 7, 8, 6, 5, 3, 4, 3, 1, 3, 0, 6, 8, 5, 7, 8, 2, 8, 8, 0, 1, 8, 9, 9, 0, 3, 9, 8, 0, 4, 2, 9, 4, 5, 3, 5, 7, 9, 5, 3, 4, 1, 5, 3, 8, 0, 4, 3, 7, 1, 4, 8, 9, 6, 8, 8, 5, 3, 3, 7, 1, 2, 9, 9, 2, 1, 5, 8, 5, 4, 4, 8, 5, 2, 1, 8, 9, 9

Offset: 0

Vaclav Kotesovec, May 29 2015

			0.16182024229485656180261334985786534313068578288018990398...

Vaclav Kotesovec, The integration of q-series

Cf. A000727, A258232, A258406, A258407, A258404, A258405.

evalf(Sum((2*Pi*(-1)^m / cosh(sqrt(7 - 4*m + 12*m^2)*Pi/2)), m=-infinity..infinity), 120); # Vaclav Kotesovec, Dec 04 2015

nmax=200; p=1; q4=Table[PrintTemporary[n]; p=Expand[p*(1-x^n)^4]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]],{n,1,nmax}]; q4n=N[q4,1000]; Table[SequenceLimit[Take[q4n,j]],{j,Length[q4n]-100,Length[q4n],10}]
NSum[2*(-1)^m*Pi/Cosh[Sqrt[7 - 4*m + 12*m^2]*Pi/2], {m, -Infinity, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100] (* Vaclav Kotesovec, Dec 04 2015 *)
RealDigits[NIntegrate[QPochhammer[x]^4, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

default(realprecision, 93);
b(n) = cosh(sqrt(7 - 4*n + 12*n^2)*Pi/2);
2*Pi*(1/b(0) + sumalt(n=1, (-1)^n*(1/b(n) + 1/b(-n)))) \\ Gheorghe Coserea, Sep 26 2018

Sum_{m = -infinity..infinity} (2*Pi*(-1)^m / cosh(sqrt(7 - 4*m + 12*m^2)*Pi/2)). - Vaclav Kotesovec, Dec 04 2015

More digits from Vaclav Kotesovec, Oct 10 2023

A272198 The p-defect p - N(p) of the congruence y^2 == x^3 + 1 (mod p) for primes p, where N(p) is the number of solutions given by A272197(n).

Original entry on oeis.org

0, 0, 0, -4, 0, 2, 0, 8, 0, 0, -4, -10, 0, 8, 0, 0, 0, 14, -16, 0, -10, -4, 0, 0, 14, 0, 20, 0, 2, 0, 20, 0, 0, -16, 0, -4, 14, 8, 0, 0, 0, 26, 0, 2, 0, -28, -16, -28, 0, -22, 0, 0, 14, 0, 0, 0, 0, -28, 26, 0

Offset: 1

Wolfdieter Lang, May 02 2016

This sequence for an elliptic curve (of the Bachet-Mordell type) is discussed in the Silverman reference. In Exercise 45.5, in the table on p. 405, the p-defects are called a_p, and are shown for primes 2 to 113.
The modularity pattern series is the expansion of the 51st modular cusp form of weight 2 and level N=36, given in the table I of the Martin reference, i.e., eta^4(6*z) in powers of q = exp(2*Pi*i*z), with Im(z) > 0. Here eta is the Dedekind function. See A000727 for the expansion in powers of q^6 (after deleting a factor q^(1/6)). Note that also for the possibly bad prime 2 and the bad prime 3 this expansion gives the correct numbers 0 (the discriminant of this elliptic curve is -3^3).
See also the comment on the Martin-Ono reference in A272197 which implies that eta^4(6*z) provides the modularity sequence for this elliptic curve.
For primes p == 0 and 2 (mod 3) (A045309) a(p) = 0. The proof runs along the same line as the one given in the Silverstein reference on pp. 400 - 402 for 17 replaced by 1. From the expansion of the known modularity function eta^4(6*z) follows that only the coefficients for powers q^n with n == 1 (mod 6) are nonzero, and therefore all a(p) for primes p == 0 and 2 (mod 3) have to vanish.
If prime(n) == 1 (mod 3) = A002476(m) (for a unique m = m(n)) then prime(n) = A(m)^2 + 3*B(m)^2 with A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1. In this case (4*prime(n) - a(n)^2)/12, seems to be a square, q(m)^2. In fact is seems that (the positive) q(m) = B(m). This is true at least for the first 80 primes 1 (mod 3), i.e. for such primes <= 997. (In the Silverman reference, in hint c) for Exercise 4.5, on p. 405, a more complicated way is suggested: 4*p is decomposed there non-uniquely instead of p uniquely.) If this conjecture is true then a(n) = 2*(+/-sqrt(prime(n) - 3*B(m)^2)) = +- 2*A(m) for prime(n) = A002476(m). This leads to a bisection of the primes 1 (mod 3) into two types: type I if the + sign applies, and type II for the - sign. Primes of type I are given in A272200: 13, 19, 43, 61, 97, ... and those of type II in A272201: 7, 31, 37, 67, 73, ...

			a(1) = 2 - A272197(1) = 0, and 2 == 2(mod 3).
a(4) = 7 - A272197(4) = 7 - 11 = -4, and 7 = A002476(1) = 2^2 + 3*1^2, 2 = A001479(1+1), 7 = A272201(1), hence a(4) = -2*2 = -4.
a(6) = 13 - A272197(6) = 13 - 11 = 2, and 13 = A002476(2) = 1^2 + 3*2^2; 1 = A001479(2+1), 13 = A272200(1), hence a(6) = +2*1 = +2.

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Exercise 45.5, p. 405, Exercise 47.2, p. 415, and pp. 400 - 402 (4th ed., Pearson 2014, Exercise 5, p. 371, Exercise 2, p. 385, and pp. 366 - 368).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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, A002476, A001479, A001480, A045309, A272197, A272200, A272201.

a(n) = prime(n) - N(prime(n)), n = 1, where N(prime(n)) = A272197(n), the number of solutions of the congruence y^2 == x^3 + 1 (mod prime(n)).
a(n) = 0 for prime(n) == 0, 2 (mod 3) (see A045309).
The above given conjecture for primes 1 (mod 3) is true because Mordell proved the Ramanujan conjecture on the expansion coefficients of eta^4(6*z), and with the present a(n) the result of Ramanujan follows. See the references and a comment on A000727.
a(n) = +2*A001479(m+1) if prime(n) == A002476(m) (m is unique) is a prime of A272200 (type I).
a(n) = -2*A001479(m+1) if prime(n) == A002476(m) is from A272201 (type II).
See a comment above for the bisection of the primes 1 (mod 3) into type I and II.

A000730 Expansion of Product_{n>=1} (1 - x^n)^7.

Original entry on oeis.org

1, -7, 14, 7, -49, 21, 35, 41, -49, -133, 98, -21, 126, 112, -176, -105, -126, 140, -35, 147, 259, 98, -420, -224, 238, -455, 273, -14, 322, 406, -35, -7, -637, -196, 245, -181, -574, 462, 147, 924, 217, -329, -140, -7, -371, -777

Offset: 0

N. J. A. Sloane

sign

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

CoefficientList[QPochhammer[x]^7 + O[x]^50, x] (* Jean-François Alcover, Feb 10 2016 *)

a(0) = 1, a(n) = -(7/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017

G.f.: exp(-7*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

A153728 Expansion of q^(-1/3) * (eta(q)^8 + 8 * eta(q^4)^8) in powers of q^2.

Original entry on oeis.org

1, 20, -70, 56, -125, 308, 110, -520, 57, 0, 182, -880, 1190, 884, 0, -1400, -1330, 1820, -646, 0, -1331, 380, 1120, 2576, 0, 1748, -3850, -3400, 2703, -2500, 3458, 0, -1150, -5236, 0, 6032, 6160, -3220, 4466, 0, -7378, -3920, 0, 2200, 0, 812, -4030, 5600, -4913

Offset: 0

Michael Somos, Dec 31 2008

sign

This is a member of an infinite family of integer weight modular forms. g_1 = A097195, g_2 = A000727, g_3 = A152243, g_4 = A153728. - Michael Somos, Jun 10 2015

			G.f. = 1 + 20*x - 70*x^2 + 56*x^3 - 125*x^4 + 308*x^5 + 110*x^6 - 520*x^7 + ...
G.f. = q + 20*q^7 - 70*q^13 + 56*q^19 - 125*q^25 + 308*q^31 + 110*q^37 + ...

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

Cf. A000727, A000731, A097195, A152243, A153729, A161969.

A := Basis( CuspForms( Gamma0(36), 4), 289); A[1] + 20*A[7] - 70*A[12]; /* Michael Somos, Jun 10 2015 */

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

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

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, if( e%2, 0, (-p)^(3*e/2)), for(x=1, sqrtint(p\3), if( issquare(p-3*x^2, &y), break)); if( y%3!=1, y=-y); y*=2; y = y^3 - 3*p*y; a0=1; a1=y; for(i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 10 2015 */

a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-1)^(e/2) * p^(3*e/2) if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^3 if p == 1 (mod 6) where b(p) = (x^2 - 3*p)*x, 4*p = x^2 + 3*y^2, |x| < |y| and x == 2 (mod 3).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 648 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A153729.
a(n) = A000731(2*n) = A153729(2*n) = A161969(2*n). - Michael Somos, Jun 10 2015

A010817 Expansion of Product_{k>=1} (1 - x^k)^9.

Original entry on oeis.org

1, -9, 27, -12, -90, 135, 54, -99, -189, -85, 657, -162, -135, -171, -810, 702, 495, 837, -673, -900, 243, -1053, -297, 1566, 2700, -1764, 81, -1188, -1377, 270, -2043, 3321, -756, 3726, 3015, -4563, -3348, 504, -351, -1350, -468

Offset: 0

N. J. A. Sloane

sign

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000203.

# DedekindEta is defined in A000594.
A010817List(len) = DedekindEta(len, 9)
A010817List(41) |> println # Peter Luschny, Mar 10 2018

a(0) = 1, a(n) = -(9/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017

G.f.: exp(-9*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

A272197 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, 11, 11, 11, 17, 11, 23, 29, 35, 47, 41, 35, 47, 53, 59, 47, 83, 71, 83, 83, 83, 89, 83, 101, 83, 107, 107, 113, 107, 131, 137, 155, 149, 155, 143, 155, 167, 173, 179, 155, 191, 191, 197, 227, 227, 251, 227, 251, 233, 239, 227, 251, 257, 263, 269, 299, 251, 281

Offset: 1

Wolfdieter Lang, May 02 2016

This elliptic curve is discussed in the Silverman reference. In the table the p-defects prime(n) - a(n) are shown for primes 2 to 113.
In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the eighth row, starting with Conductor 36, as a strong Weil curve for the weight 2 newform eta(6*z)^4, with Im(z) > 0, and the Dedekind eta function. See A000727 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z/3)*eta(z)^4. For the q-expansion of eta(6*z)^4 one has 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.

			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:  (0, 1), (0, 2), (2, 0)
3,   5,       5:  (0, 1), (0, 4), (2, 2),
                  (2, 3), (4, 0)
4,   7,      11:  (0, 1), (0, 6), (1, 3),
                  (1, 4), (2, 3), (2, 4),
                  (3, 0), (4, 3), (4, 4),
                  (5, 0), (6, 0)
5,  11,      11:  (0, 1), (0, 10), (2, 3),
                  (2, 8), (5, 4), (5, 7),
                  (7, 5), (7, 6), (9, 2),
                  (9, 9), (10, 0)

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Exercise 45.5, p. 405, Exercise 47.2, p. 415. (4th ed., Pearson 2014, Exercise 5, p. 371, Exercise 2, p. 385).

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, A000727, A272198.

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

A272200 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the first case.

Original entry on oeis.org

13, 19, 43, 61, 97, 103, 109, 127, 157, 163, 181, 193, 241, 277, 283, 331, 349, 373, 379, 409, 433, 463, 487, 499, 523, 601, 607, 619, 631, 661, 673, 691, 727, 733, 757, 769, 787, 811, 859, 883, 937, 967, 991

Offset: 1

Wolfdieter Lang, Apr 28 2016

The other primes congruent to 1 modulo 3 are given in A272201.
Each prime == 1 (mod 3) has a unique representation A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1 (see also A001479). The present sequence gives such primes corresponding to A(m+1) == 1 (mod 3). The ones corresponding to A(m+1) not == 1 (mod 3) (the complement) are given in A272201.
This bisection of the primes from A002476 is needed in the formula for the coefficients of the q-expansion (q = exp(2*Pi*i*z), Im(z) > 0) of the modular cusp form (eta(6*z))^4|A000727%20which%20gives%20the%20coefficients%20of%20the%20q-expansion%20of%20F(q)%20=%20Eta64(q%5E(1/6))/q%5E(1/6)%20=%20(Product">{z=z(q)} = Eta64(q) with Dedekind's eta function. See A000727 which gives the coefficients of the q-expansion of F(q) = Eta64(q^(1/6))/q^(1/6) = (Product{m>=0} (1 - q^m))^4. The coefficients F(q) = Sum_{n>=0} f(6*n+1)*q^n are given in the Finch link on p. 5, using multiplicativity. For primes congruent to 1 modulo 6 the formula involves x_p and y_p which are the present A and B for prime p == 1 (mod 3).
See also the p-defects of the elliptic curve y^2 = x^3 + 1, related to (eta(6*z))^4, given in A272198 with another (equivalent) way to find the coefficients of the Eta64(q) expansion, hence those of F(q).

Robert Israel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

Cf. A000727, A001479, A002476, A001480, A272198, A272201 (complement relative to A002476).

filter:= proc(n) local S,x,y;
    if not isprime(n) then return false fi;
    S:= remove(hastype,[isolve(x^2+3*y^2=n)],negative);
    subs(S[1],x) mod 3 = 1
end proc:
select(filter, [seq(i,i=7..1000,6)]); # Robert Israel, Apr 29 2019

filterQ[n_] := Module[{S, x, y}, If[!PrimeQ[n], Return[False]]; S = Solve[x > 0 && y > 0 && x^2 + 3 y^2 == n, Integers]; Mod[x /. S[[1]], 3] == 1];
Select[Range[7, 1000, 6], filterQ] (* Jean-François Alcover, Apr 21 2020, after Robert Israel *)

This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) == 1 (mod 3), for m >= 1. A(m) = A001479(m+1).

A272201 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the second case.

Original entry on oeis.org

7, 31, 37, 67, 73, 79, 139, 151, 199, 211, 223, 229, 271, 307, 313, 337, 367, 397, 421, 439, 457, 541, 547, 571, 577, 613, 643, 709, 739, 751, 823, 829, 853, 877, 907, 919, 997

Offset: 1

Wolfdieter Lang, Apr 28 2016

The other primes congruent to 1 modulo 3 are given in A272200, where also more details are given.

Each prime == 1 (mod 3) has a unique representation A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1 (see also A001479). The present sequence gives these primes corresponding to A(m+1) not congruent 1 modulo 3. The ones corresponding to A(m+1) == 1 (mod 3) (the complement) are given in A272200.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000727, A001479, A002476, A001480, A272198, A272200 (complement relative to A002476).

filter:= proc(n) local S,x,y;
    if not isprime(n) then return false fi;
    S:= remove(hastype,[isolve(x^2+3*y^2=n)],negative);
    subs(S[1],x) mod 3 <> 1
end proc:
select(filter, [seq(i,i=7..1000,6)]); # Robert Israel, Apr 29 2019

filterQ[n_] := Module[{S, x, y}, If[!PrimeQ[n], Return[False]]; S = Solve[x > 0 && y > 0 && x^2 + 3 y^2 == n, Integers]; Mod[x /. S[[1]], 3] != 1];
Select[Range[7, 1000, 6], filterQ] (* Jean-François Alcover, Apr 21 2020, after Robert Israel *)

This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) not == 1 (mod 3), for m >= 1. A(m) = A001479(m+1).

A272203 P-defects p - N(p) of the congruence y^2 == x^3 - 1 (mod p) for primes p, where N(p) is the number of solutions given by A272202(n).

Original entry on oeis.org

0, 0, 0, 4, 0, 2, 0, -8, 0, 0, 4, -10, 0, -8, 0, 0, 0, 14, 16, 0, -10, 4, 0, 0, 14, 0, -20, 0, 2, 0, -20, 0, 0, 16, 0, 4, 14, -8, 0, 0, 0, 26, 0, 2, 0, 28, 16, 28, 0, -22, 0, 0, 14, 0, 0, 0, 0, 28, 26, 0, -32, 0, 16, 0, -22, 0, -32, -34, 0, 14, 0, 0, 4, 38, -8, 0, 0, -34, 0, 38, 0, -22, 0, 2, 28, 0, 0, -10, 0, -20, 0, 0, -44, 0, -32, 0, 0, 0, -8, -46, 40, 0, 0, 0, 16, -46

Offset: 1

Wolfdieter Lang, May 05 2016

The analysis of this elliptic curve runs along the same lines as in A000727, A272197 and A272198, and it is inspired by the Silverman reference where the curve y^2 = x^3 + 1 modulo primes is treated.
The series showing the modularity pattern is the expansion of the 67th modular cusp form of weight 2 and level N=144, given in the table I of the Martin reference, i.e., eta^{12}(12*z)/( eta^4(6*z)*eta^4(24*z)), symbolically 12^{12} 6^(-4) 24^{-4}, in powers of q = exp(2*Pi*i*z), with Im(z) > 0. Here eta is the Dedekind function. See A187076 for the expansion in powers of q^6 (after deleting a factor q^(1/6)). Note that also for the possibly bad prime 2 and the bad prime 3 this expansion gives the correct numbers 0 (the discriminant of this elliptic curve is -3^3).
See also the comment on the Martin-Ono reference in A272202 which implies that 12^{12} 6^(-4) 24^{-4} provides the modularity sequence for this elliptic curve.
If prime(n) == 1 (mod 3) = A002476(m) (for a unique m = m(n)) then prime(n) = A(m)^2 + 3*B(m)^2 with A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1. In this case (4*prime(n) - a(n)^2)/12, seems to be a square, q(m)^2. In fact is seems that (the positive) q(m) = B(m). If this conjecture is true then a(n) = 2*(+-sqrt(prime(n) - 3*B(m)^2)) = +- 2*A(m) for prime(n) = A002476(m). This leads to a bisection of the primes 1 (mod 3) into two types: type I if the + sign applies, and type II for the - sign. Primes of type I are given in A272204: 7,13,31,61,67, ... and those of type II in A272205: 19,37,43,73,103, ...

			a(1) = 2 - A272202(1) = 0, and 2 == 2 (mod 3).
a(4) = 7 - A272202(4) = 7 - 3 = +4, and 7 = A002476(1) = 2^2 + 3*1^2, 2 = A001479(1+1), 7 = A272204(1), hence a(4) = +2*2 = +4.
a(8) = 19 - A272202(8) = 19 - 27 = -8, and 19 = A002476(3) = 4^2 + 3*1^2; 4=A001479(3+1), 19 = A272205(1), hence a(8) = 2*(-4) = -8.

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Exercise 45.5, p. 405, Exercise 47.2, p. 415, and pp. 400 - 402 (4th ed., Pearson 2014, Exercise 5, p. 371, Exercise 2, p. 385, and pp. 366 - 368).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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, A187076, A272202, A272204, A272205.

a(n) = prime(n) - N(prime(n)), n = 1, where N(prime(n)) = A272202(n), the number of solutions of the congruence y^2 == x^3 - 1 (mod prime(n)).
a(n) = 0 for prime(n) == 0, 2 (mod 3) (see A045309).
The above given conjecture for primes 1 (mod 3) is expected to be true by analogy to the case A272198 where only the signs differ.
a(n) = +2*A001479(m+1) if prime(n) == A002476(m) (m is unique) is a prime of A272204 (type I).
a(n) = -2*A001479(m+1) if prime(n) == A002476(m) is from A272205 (type II).
See a comment above for this bisection of the primes 1 (mod 3) into type I and II.

A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, 0, -1, -3, 1, 0, 0, 2, 0, -4, 1, 0, 1, 1, 5, 2, -5, 1, 0, 0, 2, 0, 8, 5, -6, 1, 0, 1, -2, 0, -5, 10, 9, -7, 1, 0, 0, 0, -7, -4, -15, 10, 14, -8, 1, 0, 0, -2, 0, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1

Offset: 0

Peter Luschny, Oct 02 2018

The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.

			[ 0] 1,   0,   0,    0,     0,    0,     0,     0,     0,     0, ... A000007
[ 1] 1,  -1,  -1,    0,     0,    1,     0,     1,     0,     0, ... A010815
[ 2] 1,  -2,  -1,    2,     1,    2,    -2,     0,    -2,    -2, ... A002107
[ 3] 1,  -3,   0,    5,     0,    0,    -7,     0,     0,     0, ... A010816
[ 4] 1,  -4,   2,    8,    -5,   -4,   -10,     8,     9,     0, ... A000727
[ 5] 1,  -5,   5,   10,   -15,   -6,    -5,    25,    15,   -20, ... A000728
[ 6] 1,  -6,   9,   10,   -30,    0,    11,    42,     0,   -70, ... A000729
[ 7] 1,  -7,  14,    7,   -49,   21,    35,    41,   -49,  -133, ... A000730
[ 8] 1,  -8,  20,    0,   -70,   64,    56,     0,  -125,  -160, ... A000731
[ 9] 1,  -9,  27,  -12,   -90,  135,    54,   -99,  -189,   -85, ... A010817
[10] 1, -10,  35,  -30,  -105,  238,     0,  -260,  -165,   140, ... A010818
    A001489,  v , A167541, v , A319931,  v ,         diagonal: A008705
           A080956       A319930      A319932

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Vaclav Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Transpose of A286354.

Cf. A078521, A319574 (JacobiTheta3).

# DedekindEta is defined in A000594
for n in 0:10
    DedekindEta(10, n) |> println
end

DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);

eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
    return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
    print(A319933row(n, 10))

cp's OEIS Frontend

A258404 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^4 dx.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A272198 The p-defect p - N(p) of the congruence y^2 == x^3 + 1 (mod p) for primes p, where N(p) is the number of solutions given by A272197(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A000730 Expansion of Product_{n>=1} (1 - x^n)^7.

Views

Author

Keywords

References

Links

Programs

Mathematica

Formula

A153728 Expansion of q^(-1/3) * (eta(q)^8 + 8 * eta(q^4)^8) in powers of q^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A010817 Expansion of Product_{k>=1} (1 - x^k)^9.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Julia

Formula

A272197 Number of solutions of the congruence y^2 == x^3 + 1 (mod p) as p runs through the primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A272200 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the first case.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A272201 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the second case.

Views