A225923 -id:A225923 - cp's OEIS frontend

A002292 Related to representation as sums of squares.

1, 20, 74, 24, 157, 124, 478, 1480, 1198, 3044, 480, 184, 2351, 1720, 3282, 5728, 2480, 1776, 10326, 9560, 8886, 9188, 11618, 23664, 16231, 23960, 11686, 9176, 60880, 16876, 18482, 3768, 35372, 15532, 3680, 31960, 4886, 47020, 2976, 44560

Offset: 0

N. J. A. Sloane

nonn

This is A225923 without the minus signs. - Michael Somos, Aug 09 2018

J. W. L. Glaisher, On the representation of a number as sum of 2,4,6,8... squares, Quart. J. Math. 38 (1907), 1-62 (see p. 56).
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).

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

Cf. A225923.

a(n) = abs(A225923(n)). - Michael Somos, Aug 09 2018

A225912 Expansion of q * (phi(-q^2) * psi(-q)^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

0, 1, -8, 20, 0, -74, 96, -24, 0, 157, -432, 124, 0, 478, 704, -1480, 0, -1198, 792, 3044, 0, -480, -4320, 184, 0, 2351, 3344, -1720, 0, -3282, 5184, -5728, 0, 2480, -4752, 1776, 0, 10326, -6688, 9560, 0, -8886, -8448, -9188, 0, -11618, 32832, 23664, 0, -16231

Offset: 0

Michael Somos, May 20 2013

sign

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

			G.f. = q - 8*q^2 + 20*q^3 - 74*q^5 + 96*q^6 - 24*q^7 + 157*q^9 - 432*q^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A225923 (bisection?)

a[ n_] := SeriesCoefficient[ -(EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, I q^(1/2)]^2 / 4 )^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q]^2 QPochhammer[ q^4])^4, {q, 0, n}];

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

Expansion of (eta(q)^2 * eta(q^4))^4 in powers of q.
Euler transform of period 4 sequence [-8, -8, -8, -12, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^14 (t/i)^6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A225872.
G.f.: x * (Product_{k>0} (1 - x^k)^2 * (1 - x^(4*k)))^4.

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A002292 Related to representation as sums of squares.

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A225912 Expansion of q * (phi(-q^2) * psi(-q)^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions.

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