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A002107 Expansion of Product_{k>=1} (1 - x^k)^2.
(Formerly M0091 N0028)
16
1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, 0, 0, 2, 3, -2, 2, 0, 0, -2, -2, 0, 0, -2, -1, 0, 2, 2, -2, 2, 1, 2, 0, 2, -2, -2, 2, 0, -2, 0, -4, 0, 0, 0, 1, -2, 0, 0, 2, 0, 2, 2, 1, -2, 0, 2, 2, 0, 0, -2, 0, -2, 0, -2, 2, 0, -4, 0, 0, -2, -1, 2, 0, 2, 0, 0, 0, -2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*. The even partitions number 5 and the odd partitions number 4, so a(4)=5-4=1. - Jon Perry, Apr 04 2004

Also, number of partitions of n into parts of -2 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004

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

Number 68 of the 74 eta-quotients listed in Table I of Martin 1996.

REFERENCES

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).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

G. E. Andrews, Advanced problems 6562, Amer. Math. Monthly 94, 1987.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)

S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.

J. T. Joichi, Hecke-Rogers, Andrews identities; combinatorial proofs, Discrete Mathematics, Vol. 84, Issue 3, 1990, pp. 255-259.

Victor G. Kac and Dale H. Peterson, Infinite-Dimensional Lie Algebras, Theta Functions and Modular Forms, Advances in Mathematics (1984), 53. 125-264, see page 261, (5.19).

Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449, 2018.

M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

V. Kotesovec, The integration of q-series

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.

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

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

FORMULA

Expansion of q^(-1/12) * eta(q)^2 in powers of q. - Michael Somos, Mar 06 2012

Euler transform of period 1 sequence [ -2, ...]. - Michael Somos, Mar 06 2012

a(n) = b(12*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 5 (mod 12), b(p^e) = (e + 1) * (-1)^(e*x) if p == 1 (mod 12) where p = x^2 + 9*y^2. - Michael Somos, Sep 16 2006

Convolution inverse of A000712.

a(0) = 1, a(n) = 1/n*sum(k=0,n-1, -2*a(k)*sigma_1(n-k)). - Joerg Arndt, Feb 05 2011

Expansion of f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, May 17 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12 (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, May 17 2015

a(n) = Sum_{k=0..n} A010815(k)*A010815(n-k); self convolution of A010815. - Gevorg Hmayakyan, Sep 18 2016

G.f.: Sum_{m, n in Z, n >= 2*|m|} (-1)^n * x^((3*(2*n + 1)^2 - (6*m + 1)^2)/24). - Seiichi Manyama, Oct 01 2016

G.f.: exp(-2*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

EXAMPLE

G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^5 - 2*x^6 - 2*x^8 - 2*x^9 + x^10 + ...

G.f. = q - 2*q^13 - q^25 + 2*q^37 + q^49 + 2*q^61 - 2*q^73 - 2*q^97 - 2*q^109 + ...

MATHEMATICA

terms = 78; Clear[s]; s[n_] := s[n] = Product[(1 - x^k)^2, {k, 1, n}] // Expand // CoefficientList[#, x]& // Take[#, terms]&; s[n = 10]; s[n = 2*n]; While[s[n] != s[n - 1], n = 2*n]; A002107 = s[n] (* Jean-François Alcover, Jan 17 2013 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^2, {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

PROG

(PARI) {a(n) = my(A, p, e, x); if( n<0, 0, n = 12*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%12>1, if( e%2, 0, (-1)^((p%12==5) * e/2)), for( i=1, sqrtint(p\9), if( issquare(p - 9*i^2), x=i; break)); (e + 1) * (-1)^(e*x))))}; /* Michael Somos, Aug 30 2006 */

(PARI) {a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^2, n))}; /* Michael Somos, Aug 30 2006 */

(PARI) Vec(eta(x)^2) \\ Charles R Greathouse IV, Apr 22 2016

(MAGMA) Basis( CuspForms( Gamma1(144), 1), 926) [1]; /* Michael Somos, May 17 2015 */

(Julia) # DedekindEta is defined in A000594.

A002107List(len) = DedekindEta(len, 2)

A002107List(78) |> println # Peter Luschny, Mar 09 2018

CROSSREFS

Cf. A000712 (reciprocal of g.f.), A010815, A258406.

Sequence in context: A318658 A318512 A295310 * A208845 A232506 A133099

Adjacent sequences:  A002104 A002105 A002106 * A002108 A002109 A002110

KEYWORD

sign,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 17 16:44 EST 2018. Contains 317276 sequences. (Running on oeis4.)