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A105782 Coefficients of the C-Rogers mod 14 identity. 3
1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 144, 179, 222, 275, 338, 414, 507, 617, 748, 906, 1093, 1314, 1578, 1888, 2253, 2685, 3190, 3782, 4477, 5286, 6230, 7331, 8609, 10091, 11812, 13801, 16099, 18755, 21813, 25332, 29383, 34031 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

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

G.f.: Product_{k>0} (1 - x^(14*k)) * (1 - x^(14*k - 2)) * (1 - x^(14*k - 12)) / (1 - x^k) = Sum_{k>=0} x^(k^2*+ 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^2, -x^12) / 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, 2, 12 (mod 14). - Michael Somos, Nov 21 2015

a(n) ~ sin(Pi/7) * 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 + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + 15*x^9 + ...

G.f. = q^143 + q^311 + q^479 + 2*q^647 + 3*q^815 + 4*q^983 + 6*q^1151 + 8*q^1319 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^14] QPochhammer[ x^12, x^14] QPochhammer[ x^14] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 21 2015 *)

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

CROSSREFS

Cf. A105780, A105781.

Sequence in context: A245432 A115671 A208856 * A035956 A035963 A035971

Adjacent sequences:  A105779 A105780 A105781 * A105783 A105784 A105785

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Apr 19 2005

STATUS

approved

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Last modified August 12 21:10 EDT 2020. Contains 336440 sequences. (Running on oeis4.)