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 A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q. (Formerly M1372 N0532) 24
 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 math overflow link is a conjecture that a(n) == a(9*n + 2) (mod 4). Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Number of 4-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019 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). Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558. 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 A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129] 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. H. P. Robinson, Letter to N. J. A. Sloane, Oct 07, 1976 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions 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(q) = (q * A(q^4))^4 satisfies 0 = f(B(q), B(q^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(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^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 Convolution inverse is A082304. - Michael Somos, May 16 2015 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082304. - Michael Somos, May 16 2015 Expansion of f(-x^4)^2 / f(-x)^2 = psi(x^2) / phi(-x) = psi(-x)^2 / phi(-x)^2 = psi(x)^2 / phi(-x^2)^2 = psi(x^2)^2 / psi(-x)^2 = chi(x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x)^4) = 1 / (chi(-x)^2 * chi(-x^2)^2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, May 16 2015 a(n) ~ exp(Pi*sqrt(n)) / (8*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 18 2015 EXAMPLE G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + 90*x^7 + 144*x^8 + ... G.f. = 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..40); # Alois P. Heinz, Sep 08 2008 f:=(k, M) -> mul(1-q^(k*j), j=1..M); LRBP := (L, M) -> (f(L, M)/f(1, M))^2; S := L -> seriestolist(series(LRBP(L, 80), q, 60)); S(4); # N. J. A. Sloane, Oct 20 2019 MATHEMATICA m = 38; CoefficientList[ Series[ Product[ (1 - x^(4*k))/(1 - x^k), {k, 1, m}]^2 , {x, 0, m}], x] (* Jean-François Alcover, Sep 02 2011, after g.f. *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]) / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, May 16 2015 *) a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *) a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *) a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^2, x^2])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *) 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, A082304, A098613, A127391, A127392, A210656. Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548. Sequence in context: A103577 A326508 A079006 * A246712 A279476 A281683 Adjacent sequences:  A001933 A001934 A001935 * A001937 A001938 A001939 KEYWORD nonn AUTHOR STATUS approved

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Last modified June 5 16:30 EDT 2020. Contains 334852 sequences. (Running on oeis4.)