A105780 Coefficients of the A-Rogers mod 14 identity.

1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 35, 46, 61, 79, 102, 131, 167, 211, 266, 333, 415, 515, 636, 782, 959, 1171, 1425, 1729, 2091, 2521, 3033, 3637, 4351, 5193, 6183, 7345, 8708, 10301, 12161, 14331, 16856, 19789, 23195, 27139, 31703, 36978, 43063, 50075, 58148

Offset

0,3

Comments

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

References

Agarwal, A. K., and George E. Andrews. "Rogers-Ramanujan identities for partitions with “N copies of N”." Journal of Combinatorial Theory, Series A 45.1 (1987): 40-49. See Theorem 2.

Links

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Eric Weisstein's World of Mathematics, Rogers Mod 14 Identities

Formula

Euler transform of period 14 sequence [ 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, ...]. - Michael Somos, Sep 21 2005

G.f.: Product_{k>0} (1 - x^(14*k)) * (1 - x^(14*k - 6)) * (1 - x^(14*k - 8)) / (1 - x^k) = Sum_{k>=0} x^(k^2) / (Product_{j=1..k} (1 - x^j) * (1 - x^(2*j - 1))). - Michael Somos, Sep 21 2005

Expansion of f(-x^6, -x^8) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 21 2015

Number of partitions of n into parts all not == 0, 6, 8 (mod 14). - Michael Somos, Nov 21 2015

a(n) ~ cos(Pi/14) * 11^(1/4) * exp(Pi*sqrt(11*n/21)) / (2 * 3^(1/4) * 7^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2015

Example

G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 10*x^6 + 14*x^7 + 19*x^8 + 26*x^9 + ...
G.f. = 1/q + q^167 + 2*q^335 + 3*q^503 + 5*q^671 + 7*q^839 + 10*q^1007 + 14*q^1175 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^6, x^14] QPochhammer[ x^8, x^14] QPochhammer[ x^14] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 21 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0}[[Mod[k, 14, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Nov 21 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - [ 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1][k%14 + 1] * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Sep 21 2005 */

Crossrefs

Cf. A105781, A105782.

Sequence in context: A325862 A280277 A102108 * A001522 A054405 A155167

Adjacent sequences: A105777 A105778 A105779 * A105781 A105782 A105783

Keyword

nonn

Author

Eric W. Weisstein, Apr 19 2005

Status

approved