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A124340 Number of solutions to n = x^2 + 2*y^2 + 4*(T(z) + T(w)) + 1 where x and y are integers, z and w are nonnegative integers and T(x) = (x^2+x)/2. 2
1, 2, 2, 4, 4, 4, 8, 8, 7, 8, 10, 8, 12, 16, 8, 16, 18, 14, 18, 16, 16, 20, 24, 16, 21, 24, 20, 32, 28, 16, 32, 32, 20, 36, 32, 28, 36, 36, 24, 32, 42, 32, 42, 40, 28, 48, 48, 32, 57, 42, 36, 48, 52, 40, 40, 64, 36, 56, 58, 32, 60, 64, 56, 64, 48, 40, 66 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number 18 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).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

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

Euler transform of period 8 sequence [ 2, -1, 2, -2, 2, -1, 2, -4, ...].

a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = (p^(e+1) - 1)/(p - 1) if p == 1, 7 (mod 8), a(p^e) = (p^(e+1) + (-1)^e)/(p + 1) if p == 3, 5 (mod 8).

G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).

G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(8*k))^2.

EXAMPLE

G.f. = q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 4*q^6 + 8*q^7 + 8*q^8 + 7*q^9 + ...

MATHEMATICA

a[n_] := Sum[JacobiSymbol[2, d]*n/d, {d, Divisors[n]}]; a /@ Range[80] (* Jean-Fran├žois Alcover, Jan 10 2014 *)

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^3 QPochhammer[ q^4] QPochhammer[ q^8]^2 / QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n / d * kronecker(2, d)))};

(PARI) {a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; f = kronecker( 2, p); (p^(e+1) - f^(e+1)) / (p - f)))};

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x + A)^2, n))};

CROSSREFS

Sequence in context: A033756 A320198 A320197 * A071165 A137688 A033720

Adjacent sequences:  A124337 A124338 A124339 * A124341 A124342 A124343

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Oct 26 2006

STATUS

approved

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Last modified October 22 10:24 EDT 2019. Contains 328317 sequences. (Running on oeis4.)