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A113411 Excess of number of divisors of 2n+1 of form 8k+1, 8k+3 over those of form 8k+5, 8k+7. 15
1, 2, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 1, 4, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 1, 4, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 5, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 3, 4, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 0, 2, 0, 0, 1, 6, 0, 0, 4, 2, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Bisection of A002325. Number of ways to write n as a sum of a square plus four times a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011

REFERENCES

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.55).

LINKS

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

M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

Expansion of q^(-1) * (eta(q^4)^5 * eta(q^16)^2) / (eta(q^2)^2 * eta(q^8)^3) in powers of q^2.

a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8).

Euler transform of period 8 sequence [ 2, -3, 2, 0, 2, -3, 2, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A133692. - Michael Somos, Mar 16 2011

G.f.: (Sum_{k} x^k^2) * (Sum_{k>=0} x^(2*k^2 + 2*k)).

G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>=0} F(x^(2*k + 1), x^(3*(2*k + 1))) where F(x, y) = (x + y) / (1 + x*y).

a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A112603(n). a(4*n + 1) = 2 * A033761(n).

EXAMPLE

1 + 2*x + 3*x^4 + 2*x^5 + 2*x^8 + 2*x^9 + x^12 + 4*x^13 + 4*x^16 + ...

q + 2*q^3 + 3*q^9 + 2*q^11 + 2*q^17 + 2*q^19 + q^25 + 4*q^27 + 4*q^33 + ...

MATHEMATICA

a[n_] := DivisorSum[2n+1, Switch[Mod[#, 8], 1|3, 1, 5|7, -1]&]; Table[a[n], {n, 0, 104}] (* Jean-Fran├žois Alcover, Dec 04 2015 *)

PROG

(PARI) a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, (-1)^(d%8>3)))

(PARI) a(n) = local(n1); if( n<0, 0, n1 = sqrtint(n); polcoeff( sum(k=1, n1, 2*x^k^2, 1 + x*O(x^n)) * sum(k=0, n1, x^(2*k^2 + 2*k)), n))

(PARI) a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3), n))

(PARI) a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( abs(p%8-6)==1, (1+(-1)^e)/2, e+1)))))

CROSSREFS

Cf. A033761, A112603, A133692.

Sequence in context: A174169 A248174 A125095 * A260110 A261115 A216229

Adjacent sequences:  A113408 A113409 A113410 * A113412 A113413 A113414

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 29 2005

STATUS

approved

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Last modified March 29 18:26 EDT 2020. Contains 333116 sequences. (Running on oeis4.)