A006571 -id:A006571 - cp's OEIS frontend

A124160 Erroneous version of A006571.

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

Offset: 1

dead

Barry Cipra, What's Happening in the Mathematical Sciences, Vol. 5, Amer. Math. Soc., 2002; see p. 5.

A366417 a(n) = A006571(A005117(n)).

1, -2, -1, 1, 2, -2, -2, 1, 4, 4, -1, -2, 0, 2, -2, -1, -8, 0, 2, 7, -1, 4, -2, 3, 0, -4, -8, -4, -6, 2, 8, 2, -6, 1, 0, 0, 5, 12, -14, 4, 2, -7, 1, 4, -3, 4, -6, -2, 8, -10, 16, -6, -2, 12, 0, 15, -8, -7, -16, 0, -7, 2, -4, -16, 2, 12, 18, 10, -2, -3, 9, 0, -1

Offset: 1

Mats Granvik, Oct 10 2023

sign

Cf. A006571, A366450, A005117, A002070, A023900, A191898, A366362.

nn = 73; squareFree = Select[Range[8*nn], SquareFreeQ]; b[n_] := SeriesCoefficient[q (Product[(1 - q^k), {k, 11, n, 11}] Product[1 - q^k, {k, n}])^2, {q, 0, n}]; Table[b[squareFree[[n]]], {n, 1, nn}]

a(n) = A006571(A005117(n)).

Conjecture: a(n) = A366450(A005117(n)), verified up to n = 98.

A375568 a(n) = denominator(A006571(n)/A366450(n)) if A366450(n) != 0, otherwise 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 9, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 14, 5, 1, 2, 1, 9, 1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 4, 27, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 14, 3, 5, 1, 1, 1, 1

Offset: 1

Mats Granvik, Aug 19 2024

a(n) differs from A071974 at n = 27, 32, 36, 49, 54, 72, 76, 81, 96, 98, 100, 108, 116, 125, 135, 144,...
a(n) differs from A056622 at n = 27, 32, 36, 49, 54, 72, 76, 81, 96, 98, 100, 108, 116, 125, 128, 135, 144,...
GCD(a(n), A071974(n)) differs from A071974 at n = 36, 72, 76, 100, 116, 144,...
GCD(a(n), A056622(n)) differs from A056622 at n = 36, 72, 76, 100, 116, 128, 144,...

Cf. A006571, A366450, A056622, A071974.

nn = 104; a[n_] := DivisorSum[n, MoebiusMu[#]   # &]; f = (x^3 - x^2 - y^2 - y); w[n_] := SeriesCoefficient[q*(Product[(1 - q^k), {k, 11, n, 11}]*Product[1 - q^k, {k, n}])^2, {q, 0, n}]; A006571 = ParallelTable[w[n], {n, 1, nn}]; A366450 = ParallelTable[Sum[Sum[Sum[If[GCD[f, n] == k, 1, 0]*a[k]/n, {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]; Denominator[A006571/A366450]

A002070 Coefficient of x^p (p = n-th prime) in x * Product_{k>=1} (1-x^k)^2*(1-x^11k)^2.

Original entry on oeis.org

-2, -1, 1, -2, 1, 4, -2, 0, -1, 0, 7, 3, -8, -6, 8, -6, 5, 12, -7, -3, 4, -10, -6, 15, -7, 2, -16, 18, 10, 9, 8, -18, -7, 10, -10, 2, -7, 4, -12, -6, -15, 7, 17, 4, -2, 0, 12, 19, 18, 15, 24, -30, -8, -23, -2, 14, 10, -28, -2, -18, 4, 24, 8, 12, -1, 13, 7, -22, 28, 30, -21, -20, -17, -26, -5, -1, -15, -2

Offset: 1

N. J. A. Sloane, Sep 13 2003

Form the infinite product x*[(1-x)*(1-x^11)*(1-x^2)*(1-x^22)*(1-x^3)*(1-x^33)*(1-x^4)*(1-x^44)*...]^2 and take the coefficients of x^2, x^3, x^5, x^7, x^11, x^13, x^17, x^19, ...

The primes p where A006571(p) == 0 (mod p) are called supersingular for the elliptic curve "11a3" and are given by sequence A006962. - Michael Somos, Dec 25 2010

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 = 1..10000 (first 1229 terms N. J. A. Sloane)
Goro Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 1966 209-220.
G. Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 1966 209-220. [Annotated scan of pages 218, 219 only]

Cf. A006571 (all coefficients). A006962.

a[ n_] := If[ n < 1, 0, With[ {m = Prime @ n}, SeriesCoefficient[ q (Product[ (1 - q^(11 k)), {k, Ceiling[m/11]}]Product[ 1 - q^k, {k, m}])^2, {q, 0, m}]]] (* Michael Somos, Jul 04 2011 *)

a(n) == 1 + prime(n) (mod 5) if prime(n) != 11. - Seiichi Manyama, Sep 17 2016

Conjecture: a(n) = Sum_{k=1..prime(n)} Sum_{y=1..prime(n)} Sum_{x=1..prime(n)} (A023900(k)/prime(n))[GCD(f(x,y), prime(n)) = k], where f(x,y) = x^3 - x^2 - y^2 - y. - Mats Granvik, Oct 09 2023

A060457 Number of solutions to y^2 + y = x^3 - x^2 modulo n.

Original entry on oeis.org

1, 4, 4, 8, 4, 16, 9, 16, 12, 16, 10, 32, 9, 36, 16, 32, 19, 48, 19, 32, 36, 40, 24, 64, 20, 36, 36, 72, 29, 64, 24, 64, 40, 76, 36, 96, 34, 76, 36, 64, 49, 144, 49, 80, 48, 96, 39, 128, 63, 80, 76, 72, 59, 144, 40, 144, 76, 116, 54, 128, 49, 96, 108, 128, 36, 160, 74

Offset: 1

Frank Ellermann, Apr 09 2001

Singh mistakenly called this the L-series, but the L-series for elliptic curve y^2 + y = x^3 - x^2 is A006571. - Michael Somos, Mar 20 2010

			a(5)=4 from the 4 solutions (0,0), (0,4), (1,0), (1,4) mod 5.
G.f. = x + 4*x^2 + 4*x^3 + 8*x^4 + 4*x^5 + 16*x^6 + 9*x^7 + 16*x^8 + 12*x^9 + ...

Simon Singh, Fermat's last theorem, 1997 (at the end of ch. 4).

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A061011, A061012.

{a(n) = sum(x=0, n-1, sum(y=0, n-1, (y^2 + y - x^3 + x^2) % n == 0))}; /* Michael Somos, Mar 20 2010 */

{a(n) = local(E, A, p, e); if(n<1, 0, E = ellinit( [0, -1, 1, 0, 0], 1); A = factor(n); prod( k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; (p - ellap(E, p)) * p^(e-1) )))}; /* Michael Somos, Mar 20 2010 */

More terms from Larry Reeves (larryr(AT)acm.org), Apr 13 2001

A272196 Number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod p) as p runs through the primes.

Original entry on oeis.org

2, 4, 4, 9, 10, 9, 19, 19, 24, 29, 24, 34, 49, 49, 39, 59, 54, 49, 74, 74, 69, 89, 89, 74, 104, 99, 119, 89, 99, 104, 119, 149, 144, 129, 159, 149, 164, 159, 179, 179, 194, 174, 174, 189, 199, 199, 199, 204, 209, 214, 209, 269, 249, 274, 259, 249, 259, 299, 279, 299

Offset: 1

Wolfdieter Lang, Apr 22 2016

This elliptic curve is discussed in the second Silverman reference. The present sequence is given in the rows named N_p in Table 45.6, p. 403. In the rows named a_p the p-defects prime(n) - a(n) are shown.
This sequence also gives the number of solutions of the congruences y^2 + y == x^3 - x^2 - 10*x - 20 (mod prime(n)) as well as y^2 + y == x^3 - x^2 (mod prime(n)) for n > 1 (cf. A376073). The first one is given in the Martin and Ono reference in Theorem 2, first row of the table, and the second one is given in the Frenkel reference, p. 84. (Of course, one could change the sign of y in both congruences.)
The modularity pattern for the elliptic curve y^2 = x^3 - 4*x^2 + 16 (and the ones mentioned in the previous comment and a comment below) is exhibited by the modular cusp form of weight 2 and level 11 (eta(z)*eta(11*z))^2, where eta is the Dedekind function, which in the q = exp(2*Pi*i*z), (Im(z) > 0) expansion has coefficients given in A006571 (with A006571(0) = 0). For all odd primes (2 is a bad prime), A002070(n) = A006571(prime(n)) = prime(n) - a(n), n >= 2, the p-defect. A006571(2) = -2, not 2-2 = 0. Note that the discriminant of this elliptic curve is -2^8*11 (sometimes -2^12*11 is used). Prime 11 is also bad for this curve, but A006571(11) = 1 = 11 - a(5) = 11 - 10. The curve y^2 + y = x^3 - x^2 - 10*x - 20 has discriminant -11^5 (see the first Silverman reference, pp. 46-48).
From Wolfdieter Lang, Jan 02 2017: (Start)
The congruence y^2 + y == x^3 - x^2 - 7820*x - 263580 (mod p) as p runs through the odd primes has the same number of solutions. See the Cremona link, N=11.
If b_n(Q) is the number of solutions of the Diophantine equation Q(x1,x2,x3,x4) = n with the quadratic form Q(x1,x2,x3,x4) = x1^2 + 4*(x2^2+x3^2+x4^2) + x1*x3 + 4*x2*x3 + 3*x2*x4 + 7*x3*x4 then the theta series delta(q;Q) = 1 + Sum_{n>=1} b_n(Q)*q^n equals (1/5)*E(q) + (18/5)*f(q) with the expansion coefficients of E(q) given by A185699 and those of f(q) = (eta(z)*eta(11*z))^2 with q = exp(2*Pi*i*z), (Im(z) > 0) given by A006571. See the Moreno-Wagstaff reference, pp. 245-246. b_n(Q), E(q) and f(q) are there denoted by a_n(Q), 12*E_{Chi0}(z) and f(z), respectively, and a missing n in the numerator of E_{Chi0}(z) has to be added (see A185699). (End)

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2  == x^3 - 4*x^2 + 16 (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:  (0, 0), (1, 1)
2,   3,       4:  (0, 1), (0, 2), (1, 1), (1, 2)
3,   5,       4:  (0, 1), (0, 4), (4, 1), (4, 4)
4,   7,       9:  (0, 3), (0, 4), (2, 1), (2, 6),
                  (4, 3), (4, 4), (6, 2), (6, 5)
5,  11,      10:  (0, 4), (0, 7), (4, 4), (4, 7),
                  (6, 0), (7, 3), (7, 8), (9, 5),
                  (9, 6), (10, 0)
...
--------------------------------------------------

Edward Frenkel, Liebe und Mathematik, Springer, Spektrum, 2014, p. 84.
Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York, pp. 246-247 (corrected).
J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 1986, pp. 46-48.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Table 45.6, p. 403, Theorem 47.2, p. 413 (4th ed., Pearson 2014, Table 6, p. 369, Theorem 2, p. 383)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. E. Cremona, Algorithms for Modular Elliptic Curves.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York, pp. 246-247.
J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 1986, pp. 46-48.

Cf. A002070, A006571, A060457, A185699, A376073.

a(n) gives the number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod prime(n)), n >= 1.

A030200 Expansion of q^(-1/2) * eta(q) * eta(q^11) in powers of q.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

sign

Number 52 of the 74 eta-quotients listed in Table I of Martin (1996).

In [Klein and Fricke 1892] on page 586 equation (3) first line left side has A_0 and the right side the power series r^{1/2} (1 - r - r^2 + r^5 + r^7 + ...) which is the g.f. of this sequence. A_0 and the other A_1, A_3, A_9, A_5, A_4 (in a permuted order) correspond to the nonzero 11-sections of the g.f. of this sequence. - Michael Somos, Nov 12 2014

			G.f. = 1 - x - x^2 + x^5 + x^7 - x^11 + x^13 - x^15 - x^16 - x^18 + 2*x^23 + ...
G.f. = q - q^3 - q^5 + q^11 + q^15 - q^23 + q^27 - q^31 - q^33 - q^37 + 2*q^47 +...

F. Klein and R. Fricke, Vorlesungen ueber die theorie der elliptischen modulfunctionen, Teubner, Leipzig, 1892, Vol. 2, see p. 586.
H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 203. MR1471703 (98g:14032)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A006571, A106276.

Basis( CuspForms( Gamma1(44), 1), 162) [1]; /* Michael Somos, Nov 13 2014 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^11], {x, 0, n}]; (* Michael Somos, Nov 12 2014 *)

{a(n) = if( n<0, 0, n = 2*n + 1; qfrep( [1, 0; 0, 11], n)[n] - qfrep( [3, 1; 1, 4], n)[n])}; /* Michael Somos, Nov 20 2006 */

{a(n) = my(A, p, e, f); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==11, 1, f = sum( k=0, p-1, (k^3 - k^2 - k - 1)%p == 0); if( f==0, (e-1)%3-1, if( f==1, (1 + (-1)^e) / 2, e+1)))))}; /* Michael Somos, Nov 20 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^11 + A), n))}; /* Michael Somos, Nov 20 2006 */

Euler transform of period 11 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, Nov 20 2006
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(11^e) = 1, b(p^e) = (e-1)%3 - 1 if f=0, b(p^e) = e+1 if f=3, b(p^e) = (1 + (-1)^e) / 2 if f=1 where f = number of zeros of x^3 - x^2 - x - 1 modulo p. - Michael Somos, Nov 20 2006
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(11*k)).
a(n) = sum over all solutions to x^2 + x*y + 3*y^2 = 2*n + 1 with odd integer x>0 of (-1)^y. - Michael Somos, Jan 29 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
Convolution square is A006571.

A032442 Expansion of 1 / Product_{k >= 1} (1-q^k)^2*(1-q^(11k))^2.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 754, 1169, 1780, 2685, 3996, 5894, 8600, 12450, 17860, 25442, 35964, 50519, 70490, 97800, 134892, 185099, 252664, 343280, 464200, 625033, 837998, 1119114, 1488720, 1973210, 2606028, 3430238, 4500224, 5885540

Offset: 0

N. J. A. Sloane

nonn

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 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + ...
G.f. = 1/q + 2 + 5*q + 10*q^2 + 20*q^3 + 36*q^4 + 65*q^5 + 110*q^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A006571, A028610, A128525.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^11])^-2, {x, 0, n}]; (* Michael Somos, Apr 21 2015 *)
nmax=60; CoefficientList[Series[Product[1/((1-x^k)^2 * (1-x^(11*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^-2, n))}; /* Michael Somos, Apr 21 2015 */

Expansion of 1 / (f(-x) * f(-x^11))^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Apr 21 2015
Expansion of q / eta(q)^2 * eta(q^11)^2 in powers of q. - Michael Somos, Apr 21 2015
Euler transform of period 11 sequence [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, ...]. - Michael Somos, Apr 21 2015
Given g.f. A(x), then B(q) = A(q)/q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * (w^2 + 16*v^2) - v^2 * (v + 4*u) * (w + 4*u). - Michael Somos, Apr 21 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^-1 (t/i)^-2 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 21 2015
G.f.: (Product_{k > 0} (1 - x^k)^2 * (1 - x^(11*k)))^-2.
Convolution inverse of A006571. Convolution with A028610 is A128525. - Michael Somos, Apr 21 2015
a(n) ~ exp(4*Pi*sqrt(n/11)) / (sqrt(2) * 11^(1/4) * n^(7/4)). - Vaclav Kotesovec, Oct 13 2015

A126839 Ramanujan numbers (A000594) read mod 11.

Original entry on oeis.org

1, 9, 10, 2, 1, 2, 9, 0, 9, 9, 1, 9, 4, 4, 10, 7, 9, 4, 0, 2, 2, 9, 10, 0, 7, 3, 5, 7, 0, 2, 7, 8, 10, 4, 9, 7, 3, 0, 7, 0, 3, 7, 5, 2, 9, 2, 8, 4, 8, 8, 2, 8, 5, 1, 1, 0, 0, 0, 5, 9, 1, 8, 4, 3, 4, 2, 4, 7, 1, 4, 8, 0, 4, 5, 4, 0, 9, 8, 1, 7, 1, 5, 5, 4, 9, 1, 0, 0, 4, 4, 3, 9, 4, 6, 0, 3, 4, 6, 9, 3, 2, 7, 6

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000594, this sequence (mod 11^1), A126840 (mod 11^2), A126841 (mod 11^3), A006571.

a[n_] := Mod[RamanujanTau[n], 11]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 11; \\ Amiram Eldar, Jan 05 2025

a(n) = A006571(n) (mod 11), n >= 1. For a proof see the Cowles link under A006571. See also the R. J. Mathar formula there. - Wolfdieter Lang, Feb 16 2016

A129522 Expansion of unique weight 3 level 11 multiplicative cusp form in powers of q.

Original entry on oeis.org

1, 0, -5, 4, -1, 0, 0, 0, 16, 0, -11, -20, 0, 0, 5, 16, 0, 0, 0, -4, 0, 0, 35, 0, -24, 0, -35, 0, 0, 0, -37, 0, 55, 0, 0, 64, -25, 0, 0, 0, 0, 0, 0, -44, -16, 0, 50, -80, 49, 0, 0, 0, -70, 0, 11, 0, 0, 0, 107, 20, 0, 0, 0, 64, 0, 0, 35, 0, -175, 0, -133, 0, 0

Offset: 1

Michael Somos, Apr 19 2007, Jun 06 2007

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661.

			G.f. = q - 5*q^3 + 4*q^4 - q^5 + 16*q^9 - 11*q^11 - 20*q^12 + 5*q^15 + 16*q^16 + ...

Cf. A006571, A028609, A030200, A035179, A065099, A138661.

A := Basis( CuspForms( Gamma1(11), 3), 73); A[1] - 5*A[3] + 4*A[4] - A[5]; /* Michael Somos, Mar 26 2015 */

a[ n_] := Module[ {A, B}, B = QPochhammer[ q] QPochhammer[ q^11]; A = B / (q QPochhammer[ q^3] QPochhammer[ q^33]); SeriesCoefficient[ q B^3 (1 + 3 / A) Sqrt[ q (A + 1 + 3 / A)], {q, 0, n}]]; (* Michael Somos, Mar 26 2015 *)

{a(n) = my(A, B); if( n<1, 0, n--; A = x * O(x^n); B = eta(x + A) * eta(x^11 + A); A = B /( x * eta(x^3 + A) * eta(x^33 + A)); A = B^3 * (1 + 3/A) * sqrt(x * (A + 1 + 3/A)); polcoeff(A, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==11, (-11)^e, kronecker( -11, p)==-1, if( e%2, 0, p^e), for( x=1, sqrtint(4*p\11), if( issquare(4*p - 11*x^2, &y), break)); y = y^2 - p*2; a0=1; a1=y; for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 06 2007 */

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); A = eta(x + A) * eta(x^11 + A); polcoeff( A^2 / subst(A + x * O(x^(n\2)), x, x^2) * (A^2 + 4*x * subst(A + x * O(x^(n\2)), x, x^2)^2 + 8 * x^3 * subst(A + x * O(x^(n\4)), x, x^4)^2), n))}; /* Michael Somos, Jun 06 2007 */

Expansion of (F(q)^2 + 4*F(q^2)^2 + 8*F(q^4)^2) * F(q)^2 / F(q^2) in powers of q where F(q) := eta(q) * eta(q^11) is the g.f. of A030200.
a(n) is multiplicative with a(11^e) = (-11)^e, a(p^e) = (1+(-1)^e)/2*p^e if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = a(p)*a(p^(e-1)) - p^2*a(p^(e-2)) if p == 1, 3, 4, 5, 9 (mod 11) where a(p) = y^2 - 2*p and 4*p = y^2 + 11*x^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
G.f.: (1/2) * Sum_{u, v in Z} (u*u - 3*v*v) * x^(u*u + u*v + 3*v*v). - Michael Somos, Jun 14 2007
Convolution of A006571 and A028609. - Michael Somos, Aug 14 2012
a(4*n + 2) = 0. - Michael Somos, Nov 11 2015

cp's OEIS Frontend

A124160 Erroneous version of A006571.

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A366417 a(n) = A006571(A005117(n)).

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A375568 a(n) = denominator(A006571(n)/A366450(n)) if A366450(n) != 0, otherwise 1.

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Mathematica

A002070 Coefficient of x^p (p = n-th prime) in x * Product_{k>=1} (1-x^k)^2*(1-x^11k)^2.

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A060457 Number of solutions to y^2 + y = x^3 - x^2 modulo n.

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PARI

PARI

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A272196 Number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod p) as p runs through the primes.

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A030200 Expansion of q^(-1/2) * eta(q) * eta(q^11) in powers of q.

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Magma

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PARI

PARI

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A032442 Expansion of 1 / Product_{k >= 1} (1-q^k)^2*(1-q^(11k))^2.

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