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A036016 Number of partitions of n into parts not of form 4k+2, 8k, 8k+3 or 8k-3. 4

%I #39 Mar 12 2021 22:24:42

%S 1,1,1,1,2,2,2,3,4,5,5,6,8,9,10,12,15,17,19,22,26,30,33,38,45,51,56,

%T 64,74,83,92,104,119,133,147,165,187,208,229,256,288,319,351,390,435,

%U 481,528,584,649,715,783,863,954,1047,1145,1258,1385,1517,1655,1812,1989

%N Number of partitions of n into parts not of form 4k+2, 8k, 8k+3 or 8k-3.

%C Case k=2,i=2 of Gordon/Goellnitz/Andrews Theorem.

%C Also number of partitions in which no odd part is repeated, with at most one part of size less than or equal to 2 and where differences between adjacent parts are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.

%H Seiichi Manyama, <a href="/A036016/b036016.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)

%H G. E. Andrews, <a href="http://dx.doi.org/10.1090/S0273-0979-07-01180-9">Euler's "De Partitio Numerorum"</a>, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See Th. 8.

%H S.-D. Chen and S.-S. Huang, <a href="https://doi.org/10.1142/S1793042105000030">On the series expansion of the Göllnitz-Gordon continued fraction</a>, Internat. J. Number Theory, 1 (2005), 53-63.

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.

%H Nicolas Allen Smoot, <a href="https://arxiv.org/abs/2005.09263">A Partition Function Connected with the Göllnitz--Gordon Identities</a>, arXiv:2005.09263 [math.NT], 2020. See g1(n) Table 1 p. 22.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Goellnitz-GordonIdentities.html">Goellnitz-Gordon Identities</a>

%F Expansion of f(-x^3, -x^5) / psi(-x) = psi(x^4) / f(-x, -x^7) in powers of x where phi(), f(,) are Ramanujan theta functions.

%F Euler transform of period 8 sequence [ 1, 0, 0, 1, 0, 0, 1, 0, ...]. - _Michael Somos_, Jun 28 2004

%F Let qf(a, q) = Product(1-a*q^j, j=0..infinity); g.f. is 1/(qf(q, q^8)*qf(q^4, q^8)*qf(q^7, q^8)).

%F G.f.: Sum_{k>=0} x^(k^2) Product_{i=1..k} (1 + x^(2*i - 1)) / (1 - x^(2*i)). - _Michael Somos_, Jul 24 2012

%F a(n) ~ sqrt(2+sqrt(2)) * exp(sqrt(n)*Pi/2) / (8*n^(3/4)). - _Vaclav Kotesovec_, Oct 04 2015

%e 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + 5*x^10 + ...

%p M:=100; qf:=(a,q)->mul(1-a*q^j,j=0..M); tS:=1/(qf(q,q^8)*qf(q^4,q^8)*qf(q^7,q^8)); series(%,q,M); seriestolist(%);

%t nmax=60; CoefficientList[Series[Product[1/((1-x^(8*k-1))*(1-x^(8*k-4))*(1-x^(8*k-7))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 04 2015 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - ([1, 0, 0, 1, 0, 0, 1, 0][(k-1)%8 + 1]) * x^k, 1 + x * O(x^n)), n))} /* _Michael Somos_, Jun 28 2004 */

%Y Cf. A036015, A316384.

%K nonn,easy

%O 0,5

%A _Olivier Gérard_

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)