A214129 Partitions of n into parts congruent to +-1, +-5 (mod 13).

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 17, 19, 21, 24, 27, 31, 34, 38, 42, 47, 52, 58, 64, 71, 78, 87, 95, 105, 116, 128, 140, 154, 168, 185, 202, 221, 241, 264, 287, 314, 341, 373, 405, 441, 478, 520, 564, 612, 662, 719, 777, 842

Offset: 0

Author

Michael Somos, Jul 04 2012

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
G.f. = q^-1 + q^5 + q^11 + q^17 + q^23 + 2*q^29 + 2*q^35 + 2*q^41 + 3*q^47 + 3*q^53 + ...
		

References

• Evans (1990) page 99 equation (1.10) is the connection between A214129, A214130, A214131. - Michael Somos, Nov 01 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Ronald J. Evans, Theta Function Identities, J. of Mathematical Analysis and Applications 147(1990), 97-121.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A214130, A214131.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ q, q^13] QPochhammer[ q^5, q^13] QPochhammer[ q^8, q^13] QPochhammer[ q^12, q^13]), {q, 0, n}]

PARI

{a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [ 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1][k%13 + 1] * x^k, 1 + x * O(x^n)), n))}

Formula

Expansion of f(-x^13)^2 / (f(-x, -x^12) * f(-x^5, -x^8)) in powers of x where f() is Ramanujan's two-variable theta function.

Euler transform of period 13 sequence [ 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, ...].

G.f.: 1 / (Product_{k>0} (1 - x^(13*k - 1)) * (1 - x^(13*k - 5)) * (1 - x^(13*k - 8)) * (1 - x^(13*k - 12))).

a(n) = A214130(n) + A214131(n-1).