A208856 Partitions of n into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369, 36189

Offset

0,3

Comments

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

Andrews (1987) refers to this sequence as p(T, n) where T is the set in equation (1) on page 437.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437-439.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (f(x) / f(-x) - 1) / (2 * x) in powers of x where f() is a Ramanujan theta function.

Expansion of (f(x^14, x^34) - x^4 * f(x^2, x^46)) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.

Euler transform of period 32 sequence [ 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, ...].

a(n) = A115671(n + 1). 2 * a(n) = A080054(n + 1). a(2*n) = A187154(n). a(2*n + 1) = A208851(n).

Example

1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 + 15*x^8 + 20*x^9 + ...
a(5) = 6 since  5 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 6 ways.
a(6) = 8 since  5 + 1 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 8 ways.

Mathematica

A208856[n_] := SeriesCoefficient[(1/(2*q))*((QPochhammer[-q, -q]/ QPochhammer[q, q]) - 1), {q, 0, n}]; Table[A208856[n], {n, 0, 50}] (* G. C. Greubel, Jun 19 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) - 1) / 2, n))}

Crossrefs

Cf. A080054, A115671, A187154, A208851.

Sequence in context: A224216 A245432 A115671 * A105782 A035956 A035963

Adjacent sequences: A208853 A208854 A208855 * A208857 A208858 A208859

Keyword

nonn

Author

Michael Somos, Mar 02 2012

Status

approved