A247067 -id:A247067 - cp's OEIS frontend

A030212 Glaisher's chi_4(n).

1, -4, 0, 16, -14, 0, 0, -64, 81, 56, 0, 0, -238, 0, 0, 256, 322, -324, 0, -224, 0, 0, 0, 0, -429, 952, 0, 0, 82, 0, 0, -1024, 0, -1288, 0, 1296, 2162, 0, 0, 896, -3038, 0, 0, 0, -1134, 0, 0, 0, 2401, 1716, 0, -3808, 2482, 0, 0, 0, 0, -328, 0, 0, -6958, 0, 0, 4096, 3332, 0, 0, 5152, 0, 0

Offset: 1

N. J. A. Sloane

Number 10 of the 74 eta-quotients listed in Table I of Martin (1996). Cusp form level 4 weight 5.
Called chi_4(n) by Glaisher and Hardy because as Glaisher (1907) writes on page 21 "It can be shown (see section 53) that chi_4(n) admits of an arithmetical definition, being in fact equal to one-fourth of the sum of the fourth powers of all complex numbers which have n as norm, viz. chi_4(n) = 1/4 sum_n (a + i b)^4, where a + i b is any number which has n for norm. It is in consequence of this definition that the notation chi_4(n) has been used." - Michael Somos, Jun 18 2012
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q - 4*q^2 + 16*q^4 - 14*q^5 - 64*q^8 + 81*q^9 + 56*q^10 - 238*q^13 + ...
From _Seiichi Manyama_, Apr 25 2017: (Start)
a(1) = (1 + 0i)^4 = 1,
a(2) = (1 + 1i)^4 = -4,
a(4) = (2 + 0i)^4 = 16,
a(5) = (1 + 2i)^4 + (2 + 1i)^4 = -7 - 24i - 7 + 24i = -14,
a(8) = (2 + 2i)^4 = -64,
a(9) = (3 + 0i)^4 = 81,
a(10) = (1 + 3i)^4 + (3 + 1i)^4 = 28 - 96i + 28 + 96i = 56 (End)

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)
H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 175, 4.7 Exercise 5. MR1471703 (98g:14032)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 34).
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
LMFDB, Newform orbit 4.5.b.a.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=10, called F(tau).
M. Rogers, J.G. Wan, and I. J. Zucker, Moments of Elliptic Integrals and Critical L-Values, arXiv:1303.2259 [math.NT], 2013.
Shobhit, How to prove that eta(q^4)^14/eta(q^8)^4 = 4eta(q^2)^4eta(q^4)^2eta(q^8)^4 + eta(q)^4eta(q^2)^2eta(q^4)^4?
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
Index entries for sequences mentioned by Glaisher

Cf. A002607, A215472, A247067.

Basis( CuspForms( Gamma1(4), 5), 71) [1]; /* Michael Somos, May 27 2014 */

If[SquaresR[2,#]==0,0,1/4 Plus@@((x+I y)^4/.{ToRules[Reduce[x^2+y^2==#,{x,y},Integers]]})] &/@Range[70] (* Ant King, Nov 10 2012 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^2)^2, {q, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A))^4 * eta(x^2 + A)^2, n))}; /* Michael Somos, Jul 17 2004 */

{a(n) = local(r); if( n<1, 0, r = sqrtint(n); sum( x=-r, r, sum( y=-r, r, if( x^2 + y^2 == n, (x + I*y)^4) )) / 4 )}; /* Michael Somos, Sep 12 2005 */

{a(n) = my(A, p, e, x, y, z, a0, a1); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, (-4)^e, p%4 == 3, if( e%2, 0, p^(2*e)), forstep( i=0, sqrtint(p), 2, if( issquare( p - i^2, &y), x = i; break)); a0 = 1; a1 = x = real( (x + I*y)^4 ) * 2; for( i=2, e, y = x*a1 - p^4*a0; a0=a1; a1=y); a1))) }; /* Michael Somos, Nov 18 2014 */

CuspForms( Gamma1(4), 5, prec=71).0; # Michael Somos, May 28 2013

Expansion of phi(q)^2 * psi(-q)^8 = chi(q)^6 * psi(-q)^10 = f(q)^3 * psi(-q)^7 = f(-q^2)^6 * psi(-q)^4 = f(-q^2)^10 / chi(q)^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Mar 12 2013
Expansion of eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 in powers of q.
G.f.: x * (Product_{k>0} (1 - x^k)^4 * (1 - x^(2*k))^2 * (1 - x^(4*k))^4).
G.f.: (t*t'' - 3(t')^2) / 2 where t = theta_3(x) (A000122) and t' := x * (dt/dx), t'' := (t')'. - Michael Somos, Nov 08 2005
Euler transform of period 4 sequence [ -4, -6, -4, -10, ...]. - Michael Somos, Jul 17 2004
a(n) is multiplicative with a(2^e) = (-4)^e, a(p^e) = p^(2*e) * (1 + (-1)^e)/2 if p == 3 (mod 4), a(p^e) = a(p) * a(p^(e-1)) - p^4 * a(p^(e-2)) for p == 1 (mod 4) where a(p) = 2 * Re( (x + i*y)^4 ) and p = x^2 + y^2 with even x. - Michael Somos, Nov 18 2014
Given A = A0 + A1 + A2 + A3 is the 4-section, then 0 = (A0^2 - A2^2)^2 + 4 * A0*A2*A1^2. - Michael Somos, Mar 08 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 32 (t/i)^5 f(t) where q = exp(2 Pi i t). - Michael Somos, May 28 2013
a(4*n + 3) = 0. - Michael Somos, Mar 12 2013
a(2*n) = -4 * a(n). a(4*n + 1) = A215472(n). - Michael Somos, Sep 05 2013
a(n) = 1/4 * Sum_{a^2 + b^2 = n} (a + bi)^4 = Sum_{a > 0, b >= 0, a^2 + b^2 = n} (a + bi)^4. - Seiichi Manyama, Apr 25 2017

A002607 Glaisher's chi_8(n).

Original entry on oeis.org

1, 16, 0, 256, -1054, 0, 0, 4096, 6561, -16864, 0, 0, -478, 0, 0, 65536, -63358, 104976, 0, -269824, 0, 0, 0, 0, 720291, -7648, 0, 0, -1407838, 0, 0, 1048576, 0, -1013728, 0, 1679616, 925922, 0, 0, -4317184, 3577922, 0, 0, 0, -6915294, 0, 0, 0, 5764801, 11524656, 0, -122368

Offset: 1

N. J. A. Sloane

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

			G.f. = q + 16*q^2 + 256*q^4 - 1054*q^5 + 4096*q^8 + 6561*q^9 - 16864*q^10 - ...

J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Pure and Appl. Math. 38 (1907), 289-351 (see p. 304). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]
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 (terms 1..1000 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions
W. Stein, Modular Forms Database.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences mentioned by Glaisher

Cf. A030212, A247067.

A := Basis( CuspForms( Gamma1(4), 9), 50); A[1] + 16*A[2]; /* Michael Somos, Nov 16 2014 */

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^18 (QPochhammer[ -q, q^2]^12 + 4 q / QPochhammer[ -q, q^2]^12), {q, 0, n}]; (* Michael Somos, Apr 12 2013 *)

{a(n) = local(m); if( n<1, 0, m = sqrtint(n); polcoeff( sum( j=-m, m, sum( k=-m, m, (j + I*k)^8 * x^(j^2 + k^2), x * O(x^n))) / 4, n))}; /* Michael Somos, Mar 09 2006 */

{a(n) = local(A, B); if( n<1, 0, n--; A = x * O(x^n); B = (eta(x^2 + A)^2 / eta(x + A) / eta(x^4 + A))^12; polcoeff( eta(x^2 + A)^18 * (B + 4*x / B), n))}; /* Michael Somos, Jul 25 2007 */

{a(n) = local(A, p, e, x, y, z, a0, a1); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e = A[k, 2]; if( p==2, 16^e, if( p%4 == 3, if( e%2, 0, p^(4*e)), forstep( i=0, sqrtint(p), 2, if( issquare( p - i^2, &y), x = i; break)); a0 = 1; a1 = x = real( (x + I*y)^8 ) * 2; for( i=2, e, y = x*a1 - p^8*a0; a0=a1; a1=y); a1))))) }; /* Michael Somos, Nov 18 2014 */

A = CuspForms( Gamma1(4), 9, prec=53).basis(); A[0] + 16*A[1]; # Michael Somos, Apr 12 2013

Expansion of newform of degree 1, level 4, weight 9 and nontrivial character in powers of q. - Michael Somos, Mar 09 2006
Expansion of Jacobi ((2*k(q)*k'(q))^2 + (k(q)*k'(q))^4) * (K(q) / (pi/2))^9 / 64 in powers of q. - Michael Somos, Mar 09 2006
Expansion of F(phi(q)^4, q*psi(q^2)^4) in powers of q where F(u, v) = sqrt(u) * v * (u - 16*v) * (u^2 + 4*u*v - 64*v^2) and phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 09 2006
a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = p^(4*e) * (1 + (-1)^e)/2 if p == 3 (mod 4), a(p^e) = a(p) * a(p^(e-1)) - p^8 * a(p^(e-2)) if p == 1 (mod 4) where a(p) = 2 * Re( (x + i*y)^8 ) and p = x^2 + y^2 with even x. - Michael Somos, Nov 18 2014
G.f.: (t''''*t - 28*t'''*t' + 35*t''^2) / 2 where t = phi(q) and f' := q*df/dq. - Michael Somos, Mar 09 2006
G.f.: ( Sum_{j,k} (j + i*k)^8 * x^(j^2 + k^2) ) / 4. where i^2 = -1. a(2*n) = 16*a(n). a(4*n + 3) = 0.
Expansion of q * f(-q^2)^18 * (chi(q)^12 + 4 * q / chi(q)^12) in powers of q where f(), chi() are Ramanujan theta functions. - Michael Somos, Jul 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^9 (t / i)^9 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 25 2007

Edited by Michael Somos, Mar 09 2006

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A030212 Glaisher's chi_4(n).

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A002607 Glaisher's chi_8(n).

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