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A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.
(Formerly M1372 N0532)
12
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; text; internal format)
OFFSET

0,2

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).

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.

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).

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

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.

joro, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?

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, Mar 24 2011

a(9*n + 2) = a(n) + 4 * A210656(3*n). - Michael Somos, Apr 02 2012

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 + ...

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

MATHEMATICA

m = 38; CoefficientList[ Series[ Product[ (1 - x^(4*k))/(1 - x^k), {k, 1, m}]^2 , {x, 0, m}], x] (* Jean-Fran├žois Alcover, Sep 02 2011, after g.f. *)

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))}

CROSSREFS

Cf. A001935, A079006, A098613, A127391, A127392, A210656.

Sequence in context: A185721 A103577 A079006 * A224364 A127297 A018739

Adjacent sequences:  A001933 A001934 A001935 * A001937 A001938 A001939

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 23 05:30 EDT 2014. Contains 240913 sequences.