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

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, 64, 74, 83, 92, 104, 119, 133, 147, 165, 187, 208, 229, 256, 288, 319, 351, 390, 435, 481, 528, 584, 649, 715, 783, 863, 954, 1047, 1145, 1258, 1385, 1517, 1655, 1812, 1989

Offset

0,5

Comments

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

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.

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

References

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See Th. 8.

S.-D. Chen and S.-S. Huang, On the series expansion of the Göllnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Nicolas Allen Smoot, A Partition Function Connected with the Göllnitz--Gordon Identities, arXiv:2005.09263 [math.NT], 2020. See g1(n) Table 1 p. 22.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Eric Weisstein's World of Mathematics, Goellnitz-Gordon Identities

Formula

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.

Euler transform of period 8 sequence [ 1, 0, 0, 1, 0, 0, 1, 0, ...]. - Michael Somos, Jun 28 2004

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)).

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

a(n) ~ sqrt(2+sqrt(2)) * exp(sqrt(n)*Pi/2) / (8*n^(3/4)). - Vaclav Kotesovec, Oct 04 2015

Example

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 + ...

Maple

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(%);

Mathematica

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 *)

Prog

(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 */

Crossrefs

Cf. A036015, A316384.

Sequence in context: A008906 A029074 A258741 * A051918 A365718 A163801

Adjacent sequences: A036013 A036014 A036015 * A036017 A036018 A036019

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved