login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A259285 Expansion of psi(x^2) * f(x, x^7) in powers of x where psi(), f(,) are Ramanujan theta functions. 4
1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 0, 2, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 2, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 3, 0, 1, 0, 1, 3, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 3, 1, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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

LINKS

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Number of solutions to 16*n + 13 = (8*u + 3)^2 + (8*v + 2)^2 where u,v in Z.

Euler transform of period 16 sequence [ 1, 0, 0, -1, 0, 1, 1, -2, 1, 1, 0, -1, 0, 0, 1, -2, ...].

a(9*n + 2) = A259287(n). a(9*n + 5) = a(9*n + 8) = 0.

-2 * a(n) = A134343(4*n + 3). a(n) = A000161(16*n + 13) = A025426(16*n + 13) = A025435(16*n + 13) = A025441(16*n + 13).

EXAMPLE

G.f. = 1 + x + x^2 + x^3 + x^6 + 2*x^7 + x^9 + x^10 + 2*x^12 + 2*x^13 + ...

G.f. = q^13 + q^29 + q^45 + q^61 + q^109 + 2*q^125 + q^157 + q^173 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^1, x^8] QPochhammer[ -x^2, x^8] QPochhammer[ -x^6, x^8] QPochhammer[ -x^7, x^8] QPochhammer[x^8]^2, {x, 0, n}];

a[ n_] := SeriesCoefficient[ Product[ (1 + x^(8 k - 1)) (1 + x^(8 k - 2)) (1 + x^(8 k - 6)) (1 + x^(8 k - 7)) (1 - x^(8 k))^2, {k, Ceiling[n/8]}], {x, 0, n}];

PROG

(PARI) {a(n) = my(m, s, x, c); if( n<0, 0, s = sqrtint(m = 16*n + 13); for(u = (s+3)\-8, (s-3)\8, if( issquare( m - (8*u + 3)^2, &x) && (x%8==2 || x%8==6), c++))); c};

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -1, 0, 0, 1, 0, -1, -1, 2, -1, -1, 0, 1, 0, 0, -1][k%16 + 1], 1 + x * O(x^n)), n))};

CROSSREFS

Cf. A000161, A025426, A025435, A025441, A134343, A259287.

Sequence in context: A087372 A036431 A029407 * A099544 A036414 A234954

Adjacent sequences:  A259282 A259283 A259284 * A259286 A259287 A259288

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 23 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 18 02:34 EDT 2022. Contains 356204 sequences. (Running on oeis4.)