A030210 -id:A030210 - cp's OEIS frontend

A030205 Expansion of q^(-1/2) * eta(q)^2 * eta(q^5)^2 in power of q.

1, -2, -1, 2, 1, 0, 2, 2, -6, -4, -4, 6, 1, 4, 6, -4, 0, -2, 2, -4, 6, -10, -1, -6, -3, 12, -6, 0, 8, 12, 2, 2, -2, 2, -12, -12, 2, -2, 0, 8, -11, 6, 6, -12, -6, 4, 8, 4, 2, 0, 6, 14, 4, -6, 2, -4, -6, -6, 2, -12, -11, -12, -1, 2, 20, 0, -8, -4, 18, -4, 12, 0

Offset: 0

N. J. A. Sloane

sign

This eta-quotient of conductor 20 is one of the twelve weight 2 newforms listed by Martin and Ono.
The associated elliptic curve is "20a1": y^2 = x^3 + x^2 + 4*x + 4 or "20a2": y^2 = x^3 + x^2 - x.
Number 39 of the 74 eta-quotients listed in Table I of Martin (1996).
The mentioned eta-quotient is in fact eta^2(2*z) * eta^2(10*z) with q = exp(2*Pi*i*tau) with Im(tau) > 0, I^2 = -1, with the q-expansion coefficients b(n) from the Michael Somos Oct Aug 13 2006 formula: b(2*n) = 0 and b(2*n+1) = a(n), for n >= 0. A273163(k) = b(prime(k)), k >= 1. See also the comments on multiplicativity of b(n) (called there c(n)) with b(2^k) = b(2)^k = 0, b(5^k) = b(5)^k = (-1)^k, and b(prime(n)^k) = (sqrt(prime(n)))^k*S(k,A273163(n)/sqrt(prime(n))) with Chebyshev's S polynomials (A049310), for n = 2, and n >= 4 and k >= 2. Compare this with the b(p^(e+2)) recurrence given by Michael Somos, Oct 31 2005. - Wolfdieter Lang, May 23 2016

			G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^6 + 2*x^7 - 6*x^8 - 4*x^9 - 4*x^10 + ...
G.f. of b(n) from eta^2(2*z)*eta^2(10*z) = q - 2*q^3 - q^5 + 2*q^7 + q^9 + 2*q^13 + 2*q^15 - 6*q^17 - 4*q^19 + ..., where q = exp(2*Pi*I*z).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
LMFDB, Space of modular forms of level 20 and weight 2.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)
K. Ono, Ramanujan, taxicabs, birthdates, ZIP codes and twists, Amer. Math. Monthly, 104 (1997), 912-917. MR1490909 (98i:11020)
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A030210, A049310, A159817, A273163.

Basis( CuspForms( Gamma0(20), 2), 145) [1]; /* Michael Somos, May 27 2014 */

A := Basis( CuspForms( Gamma1(20), 2), 145); A[1] - 2*A[3]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^5])^2, {x, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = if( n<0, 0, n = 2*n + 1; ellan( ellinit( [0, 1, 0, 4, 4], 1), n)[n])}; /* Michael Somos, Oct 31 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^5 + A))^2, n))}; /* Michael Somos, Oct 31 2005 */

{a(n) = if( n<0, 0, n = 2*n + 1; ellan( ellinit( [0, 1, 0, -1, 0], 1), n)[n])}; /* Michael Somos, Aug 13 2006 */

{a(n) = my(A, p, e, x, y, a0, a1); 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==5, (-1)^e, a0=1; a1 = y = -sum( x=0, p-1, kronecker( x^3 + x^2 - x, p)); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 13 2006 */

CuspForms( Gamma0(20), 2, prec=92).0; # Michael Somos, May 28 2013

Euler transform of period 5 sequence [ -2, -2, -2, -2, -4, ...]. - Michael Somos, Oct 31 2005
Given g.f. A(x), then B(x) = q * A(q)^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u*w * (u + 8*v + 16*w) - v^3. - Michael Somos, Oct 31 2005
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, else b(p^(e+2)) = b(p)*b(p^(e+1)) - p*b(p^e). - Michael Somos, Oct 31 2005
a(n) = b(2*n + 1) and b(p) = p minus number of points of elliptic curve "20a1" or "20a2" modulo p. - Michael Somos, Aug 13 2006
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(5*k)))^2.
a(121*n + 60) = -11 * a(n).
Convolution square is A030210. - Michael Somos, Jun 13 2014
a(n) = (-1)^n * A159817(n). - Michael Somos, Jun 10 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 20 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 10 2015

A030202 Expansion of q^(-1/4) * eta(q) * eta(q^5) in powers of q.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Number 62 of the 74 eta-quotients listed in Table I of Martin (1996).

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

			G.f. = 1 - x - x^2 + x^6 + 2*x^7 - 2*x^10 + x^11 - x^12 - 2*x^15 + x^20 + ...
G.f. = q - q^5 - q^9 + q^25 + 2*q^29 - 2*q^41 + q^45 - q^49 - 2*q^61 + q^81 + ...

Bruce Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; see page 44.

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
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A030205, A030210, A159818.

Basis( CuspForms( Gamma1(80), 1), 413)[1]; /* Michael Somos, May 16 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^5], {x, 0, n}] (* Michael Somos, Aug 08 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 1, Pi/5, q^2] EllipticTheta[ 1, 2 Pi/5, q^2] / Sqrt[5], {q, 0, 4 n + 1}] // FullSimplify; (* Michael Somos, Aug 08 2011 *)

{a(n) = if( n<0, 0, polcoeff( eta(x^5 + x * O(x^n)) * eta(x + x * O(x^n)), n))}; /* Michael Somos, Sep 04 2007 */

{a(n) = my(A, p, e, x, y); 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==5, (-1)^e, p%20>10, !(e%2), p%4==3, kronecker( -4, e+1), for( y=1, sqrtint(p\5), if( issquare(p - 5*y^2), x=y; break)); (-1)^(e*x) * (e+1))))}; /* Michael Somos, Sep 04 2007 */

Expansion of f(-x, -x^4) * f(-x^2, -x^3) in powers of x where f() is the Ramanujan two-variable theta function.
Expansion of q^(-1) * (phi(q) * phi(q^20) - phi(q^4) * phi(q^5)) / 2 in powers of q^4 where phi() is a Ramanujan theta function.
Euler transform of period 5 sequence [ -1, -1, -1, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 80^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, b(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), b(p^e) = (i^n +(-i)^n)/2 if p == 3, 7 (mod 20), b(p^e) = (-1)^(e*y) * (e+1) if p == 1, 9 (mod 20) where p = x^2 + 5*y^2. - Michael Somos, Sep 04 2007
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(5*k)).
a(5*n + 3) = a(5*n + 4) = a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -a(n). - Michael Somos, May 16 2015
Convolution square is A030205. - Michael Somos, May 16 2015
a(n) = (-1)^n * A159818(n). - Michael Somos, May 16 2015

A143462 Expansion of 1/(1 + 4x + 8x^2).

Original entry on oeis.org

1, -4, 8, 0, -64, 256, -512, 0, 4096, -16384, 32768, 0, -262144, 1048576, -2097152, 0, 16777216, -67108864, 134217728, 0, -1073741824, 4294967296, -8589934592, 0, 68719476736, -274877906944, 549755813888, 0, -4398046511104, 17592186044416

Offset: 0

Michael Somos, Aug 16 2008

			1 - 4*x + 8*x^2 - 64*x^4 + 256*x^5 - 512*x^6 + 4096*x^8 - 16384*x^9 + ...

Index entries for linear recurrences with constant coefficients, signature (-4,-8).

A030210(2^n) = 2^n * A108520(n) = a(n).

I:=[1,-4]; [n le 2 select I[n] else -4 * Self(n-1) - 8 * Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

A143462 := n -> `if`(n=0, 1, (-4)^n*hypergeom([1/2-n/2, -n/2], [-n], 2)):
seq(simplify(A143462(n)), n=0..29); # Peter Luschny, Dec 17 2015

CoefficientList[Series[1/(1 + 4*x + 8*x^2), {x, 0, 30}], x] (* Jinyuan Wang, Mar 10 2020 *)

{a(n) = (-64)^(n \ 4) * [1, -4, 8, 0][n%4 + 1]}

{a(n) = n--; -2 * 2^n * ((-1 + I)^n + (-1 - I)^n)}

{a(n) = n--; simplify( -4 * (2 * quadgen(8))^n * polchebyshev(n, 1, -1 / quadgen(8)))}

G.f.: 1/(1 + 4*x + 8*x^2).
E.g.f.: (cos(2*x) - sin(2*x))/exp(2*x).
a(n) = -4*a(n-1) - 8*a(n-2).
a(n+4) = -64*a(n).
G.f.: 1/(1 + 4*x/(1 - 2*x/(1 + 2*x))) = 1 - 4*x/(1 + 2*x/(1 - 2*x/(1 + 4*x))). - Michael Somos, Jan 03 2013
a(n) = (-4)^n*hypergeom([1/2-n/2, -n/2], [-n], 2) for n >= 1. - Peter Luschny, Dec 17 2015

A346193 Convolution of level 5 of the divisor function.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 4, 7, 6, 15, 17, 27, 34, 36, 52, 64, 75, 91, 102, 122, 155, 169, 193, 228, 263, 276, 326, 349, 415, 430, 500, 520, 620, 681, 727, 741, 881, 880, 1090, 1020, 1192, 1178, 1375, 1513, 1590, 1557, 1809, 1756, 2274, 2024, 2323, 2245, 2626, 2865

Offset: 1

Amiram Eldar, Jul 09 2021

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Şaban Alaca and Kenneth S. Williams, Evaluation of the convolution sums ..., Journal of Number Theory, Vol. 124, No. 2 (2007) pp. 491-510.
Mathieu Lemire and Kenneth S. Williams, Evaluation of two convolution sums involving the sum of divisors function, Bulletin of the Australian Mathematical Society, Vol. 73, No. 1 (2006), pp. 107-115.
Emmanuel Royer, Evaluating convolution sums of the divisor function by quasimodular forms, International Journal of Number Theory, Vol. 3, No. 2 (2007) pp. 231-261.

Cf. A000203, A000385, A001158, A030210, A218276, A218277, A218278.

a[n_] := Sum[DivisorSigma[1, k] * DivisorSigma[1, n - 5*k], {k, 1, (n - 1)/5}]; Array[a, 100]
(* or *)
c[n_] := SeriesCoefficient[q * (QPochhammer[q] * QPochhammer[q^5])^4, {q, 0, n}]; a[n_] := 5 * DivisorSigma[3, n]/312 + If[Divisible[n, 5], 125 * DivisorSigma[3, n/5]/312, 0] - n * DivisorSigma[1, n]/20 - If[Divisible[n, 5], n * DivisorSigma[1, n/5]/4, 0] + DivisorSigma[1, n]/24 + If[Divisible[n, 5], DivisorSigma[1, n/5]/24, 0] - c[n]/130; Array[a, 100]

a(n) = Sum_{k < n/5} sigma(k) * sigma(n-5*k).

a(n) = 5*sigma_3(n)/312 + 125*sigma_3(n/5)/312 + (1/24 - n/20)*sigma(n) + (1/24 - n/4)*sigma(n/5) - c_5(n)/130, where sigma_3(n/5) = sigma(n/5) = 0 if n is not divisible by 5, and c_5(n) is the coefficient of q^n in the expansion of (eta(q) * eta(q^5))^4 (A030210).

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A030205 Expansion of q^(-1/2) * eta(q)^2 * eta(q^5)^2 in power of q.

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A030202 Expansion of q^(-1/4) * eta(q) * eta(q^5) in powers of q.

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A143462 Expansion of 1/(1 + 4*x + 8*x^2).

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A346193 Convolution of level 5 of the divisor function.

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A143462 Expansion of 1/(1 + 4x + 8x^2).