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A001936
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Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.
(Formerly M1372 N0532)
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11
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1, 2, 5, 10, 18, 32, 55, 90, 144, 226, 346, 522, 777, 1138, 1648, 2362, 3348, 4704, 6554, 9056, 12425, 16932, 22922, 30848, 41282, 54946, 72768, 95914, 125842, 164402, 213901, 277204, 357904, 460448, 590330, 754368, 960948, 1220370, 1545306
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The Cayley reference is actually to A079006. - Michael Somos Feb 24 2011
In the mathoverflow link is the conjecture that a(n) == a(9*n + 2) (mod 4).
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REFERENCES
| A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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).
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LINKS
| joro, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?
T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
| G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....
Convolution square of A001935. A079006(n) = (-1)^n a(n).
Expansion of q^(-1/4) * (1/2) * (k / k')^(1/2) in powers of q.
Euler transform of period 4 sequence [2, 2, 2, 0, ...].
Given g.f. A(x), then B(x) = (x * A(x^4))^4 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (1 + 16*u) * (1 + 16*v) * v - u^2. - Michael Somos Jul 09 2005
Given g.f. A(x), then B(x) = x * A(x^4) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (1 + 4*u*v)^2. - Michael Somos Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - x^k))^2.
Equals A000009 convolved with A098613 - Gary W. Adamson [qntmpkt(AT)yahoo.com Mar 24 2011]
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EXAMPLE
| 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + 90*x^7 + 144*x^8 + ...
q + 2*q^5 + 5*q^9 + 10*q^13 + 18*q^17 + 32*q^21 + 55*q^25 + 90*q^29 + ...
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MAPLE
| with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr (n-> [2, 2, 2, 0] [modp(n-1, 4)+1]): seq (a(n), n=0..38); # Alois P. Heinz, Sep 08 2008
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MATHEMATICA
| m = 38; CoefficientList[ Series[ Product[ (1 - x^(4*k))/(1 - x^k), {k, 1, m}]^2 , {x, 0, m}], x] (* From Jean-François Alcover, Sep 02 2011, after g.f. *)
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PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( (eta(x^4 + x * O(x^n)) / eta(x + x * O(x^n)))^2, n))}
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 / if(k%4, 1 - x^k, 1), 1 + x * O(x^n))^2, n))}
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CROSSREFS
| Cf. A001935, A079006, A098613, A127391, A127392.
Sequence in context: A185721 A103577 A079006 * A127297 A018739 A011893
Adjacent sequences: A001933 A001934 A001935 * A001937 A001938 A001939
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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