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%I #27 Mar 12 2021 22:24:43
%S 1,1,1,2,3,4,6,8,11,15,20,26,34,44,56,72,91,114,144,179,222,275,338,
%T 414,507,617,748,906,1093,1314,1578,1888,2253,2685,3190,3782,4477,
%U 5286,6230,7331,8609,10091,11812,13801,16099,18755,21813,25332,29383,34031
%N Coefficients of the C-Rogers mod 14 identity.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A105782/b105782.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RogersMod14Identities.html">Rogers Mod 14 Identities</a>
%F 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
%F 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
%F 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
%F Number of partitions of n into parts all not == 0, 2, 12 (mod 14). - _Michael Somos_, Nov 21 2015
%F 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
%e 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 + ...
%e G.f. = q^143 + q^311 + q^479 + 2*q^647 + 3*q^815 + 4*q^983 + 6*q^1151 + 8*q^1319 + ...
%t 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 *)
%t 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 *)
%o (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 */
%Y Cf. A105780, A105781.
%K nonn
%O 0,4
%A _Eric W. Weisstein_, Apr 19 2005
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