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A022567
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Expansion of Product (1 + q^m)^2; m=1..inf.
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5
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1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 128, 176, 238, 319, 426, 562, 736, 960, 1242, 1598, 2048, 2608, 3306, 4175, 5248, 6570, 8198, 10190, 12622, 15589, 19190, 23552, 28830, 35190, 42842, 52034, 63040, 76198, 91904, 110604, 132832, 159216, 190464, 227417
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of n into 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*, thus a(4)=9 - Jon Perry (perry(AT)globalnet.co.uk), Apr 04 2004
Number of partitions of n into odd parts, each part being of two kinds. E.g. a(3)=6 because we have 3, 3', 1+1+1, 1+1+1', 1+1'+1', 1'+1'+1'. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2005
Euler transform of period 2 sequence [2,0,2,0,...]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2005
Equals A000041 convolved with A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]
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REFERENCES
| Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, Arxiv preprint arXiv:1110.5103, 2011
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 852
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FORMULA
| a(n) = p(n)+p(n-1)+p(n-3)+p(n-6)+...+p(n-k*(k+1)/2)+..., where p() is A000041(). E.g. a(8) = p(8)+p(7)+p(5)+p(2) = 22+15+7+2 = 46. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 09 2004
Expansion of q^(-1/12) * (eta(q^2) / eta(q))^2 in powers of q.
Expansion of chi(-q)^(-2) in powers of q where chi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (1/2) / f(t) where q = exp(2 pi i t).
G.f.: Product_{k>0} (1 + x^k)^2.
Convolution square of A000009. Convolution inverse of A022597.
Parity result: a(n) is even except when n is twice a generalised pentagonal number (i.e. of the form 2*A001318(m) for some m). [From Peter Bala (pbala(AT)talktalk.net), Mar 19 2009]
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EXAMPLE
| q + 2*q^13 + 3*q^25 + 6*q^37 + 9*q^49 + 14*q^61 + 22*q^73 + 32*q^85 + ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-2 , {q, 0, n}] (* Michael Somos Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^2, {q, 0, n}] (* Michael Somos Jul 11 2011 *)
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PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^2, n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^2, n))}
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CROSSREFS
| Cf. A000009, A022597.
A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]
Sequence in context: A173303 A058609 A128518 * A134004 A123631 A018060
Adjacent sequences: A022564 A022565 A022566 * A022568 A022569 A022570
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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