A105781 Coefficients of the B-Rogers mod 14 identity.

1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 30, 40, 53, 68, 88, 113, 143, 181, 228, 284, 354, 439, 541, 665, 815, 993, 1208, 1465, 1769, 2132, 2563, 3070, 3671, 4379, 5209, 6185, 7329, 8663, 10223, 12041, 14153, 16609, 19459, 22755, 26571, 30979, 36059, 41915, 48654

Offset

0,3

Comments

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

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, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, ...]. - Michael Somos, Sep 21 2005

G.f.: Product_{k>0} (1 - x^(14*k)) * (1 - x^(14*k - 4)) * (1 - x^(14*k - 10)) / (1 - x^k) = Sum_{k>=0} x^(k^2 + k) / ((1 - x^(2*k + 1)) * Product_{j=1..k} (1 - x^j) * (1 - x^(2*j - 1))). - Michael Somos, Sep 21 2005

Expansion of f(-x^4, -x^10) / 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, 4, 10 (mod 14). - Michael Somos, Nov 21 2015

a(n) ~ 11^(1/4) * cos(3*Pi/14) * 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 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 17*x^8 + 23*x^29 + ...
G.f. = q^47 + q^215 + 2*q^383 + 3*q^551 + 4*q^719 + 6*q^887 + 9*q^1055 + 12*q^1223 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^4, x^14] QPochhammer[ x^10, 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, 0, 1, 1, 1, 1, 1, 0, 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, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1][k%14 + 1] * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Sep 21 2005 */

Crossrefs

Cf. A105780, A105782.

Sequence in context: A248475 A035952 A335754 * A035958 A035965 A035973

Adjacent sequences: A105778 A105779 A105780 * A105782 A105783 A105784

Keyword

nonn

Author

Eric W. Weisstein, Apr 19 2005

Status

approved