cp's OEIS Frontend

A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.

%I A001936 M1372 N0532 #86 Feb 16 2025 08:32:24
%S A001936 1,2,5,10,18,32,55,90,144,226,346,522,777,1138,1648,2362,3348,4704,
%T A001936 6554,9056,12425,16932,22922,30848,41282,54946,72768,95914,125842,
%U A001936 164402,213901,277204,357904,460448,590330,754368,960948,1220370,1545306
%N A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.
%C A001936 The Cayley reference is actually to A079006. - _Michael Somos_, Feb 24 2011
%C A001936 In the math overflow link is a conjecture that a(n) == a(9*n + 2) (mod 4).
%C A001936 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A001936 Number of 4-regular bipartitions of n. - _N. J. A. Sloane_, Oct 20 2019
%C A001936 The g.f. in the form A(x) = Sum_{k >= 0} x^(k*(k+1)) / (1 + 2*Sum_{k >= 1} (-1)^k * x^(k^2)) == Sum_{k >= 0} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - _Peter Bala_, Jan 04 2025
%D A001936 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.
%D A001936 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
%D A001936 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001936 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001936 T. D. Noe, <a href="/A001936/b001936.txt">Table of n, a(n) for n = 0..1000</a>
%H A001936 A. Cayley, <a href="/A001934/a001934.pdf">A memoir on the transformation of elliptic functions</a>, 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 A001936 H. R. P. Ferguson, D. E. Nielsen and G. Cook, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0367322-8">A partition formula for the integer coefficients of the theta function nome</a>, Math. Comp., 29 (1975), 851-855.
%H A001936 joro on Math Overflow, <a href="http://mathoverflow.net/questions/59192/">Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?</a>.
%H A001936 T. Kathiravan and S. N. Fathima, <a href="https://doi.org/10.1007/s11139-016-9850-9">On L-regular bipartitions modulo L</a>, The Ramanujan Journal 44.3 (2017): 549-558.
%H A001936 H. P. Robinson, <a href="/A001936/a001936.pdf">Letter to N. J. A. Sloane, Oct 07, 1976</a>
%H A001936 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A001936 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A001936 G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....
%F A001936 Convolution square of A001935. A079006(n) = (-1)^n a(n).
%F A001936 Expansion of q^(-1/4) * (1/2) * (k / k')^(1/2) in powers of q.
%F A001936 Euler transform of period 4 sequence [ 2, 2, 2, 0, ...].
%F A001936 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
%F A001936 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
%F A001936 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.
%F A001936 Equals A000009 convolved with A098613. - _Gary W. Adamson_, Mar 24 2011
%F A001936 a(9*n + 2) = a(n) + 4 * A210656(3*n). - _Michael Somos_, Apr 02 2012
%F A001936 Convolution inverse is A082304. - _Michael Somos_, May 16 2015
%F A001936 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
%F A001936 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
%F A001936 a(n) ~ exp(Pi*sqrt(n)) / (8*sqrt(2)*n^(3/4)). - _Vaclav Kotesovec_, Aug 18 2015
%F A001936 G.f.: A(x) = Sum_{n >= 0} x^(n*(n+1)) / Sum_{n = -oo..oo} (-1)^n*x^(n^2). - _Peter Bala_, Feb 19 2021
%e A001936 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 + ...
%e A001936 G.f. = q + 2*q^5 + 5*q^9 + 10*q^13 + 18*q^17 + 32*q^21 + 55*q^25 + 90*q^29 + ...
%p A001936 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
%p A001936 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
%t A001936 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. *)
%t A001936 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]) / (2 x^(1/4)), {x, 0, n}]; (* _Michael Somos_, May 16 2015 *)
%t A001936 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 *)
%t A001936 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* _Michael Somos_, May 16 2015 *)
%t A001936 a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^2, x^2])^2, {x, 0, n}]; (* _Michael Somos_, May 16 2015 *)
%o A001936 (PARI) {a(n) = if( n<0, 0, polcoeff( (eta(x^4 + x * O(x^n)) / eta(x + x * O(x^n)))^2, n))};
%o A001936 (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))};
%Y A001936 Cf. A001935, A022568, A079006, A082304, A098613, A127391, A127392, A210656.
%Y A001936 Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548, A333374.
%K A001936 nonn,easy
%O A001936 0,2
%A A001936 _N. J. A. Sloane_