|
%I #24 Mar 12 2021 22:24:43
%S 1,1,2,3,5,7,10,14,19,26,35,46,61,79,102,131,167,211,266,333,415,515,
%T 636,782,959,1171,1425,1729,2091,2521,3033,3637,4351,5193,6183,7345,
%U 8708,10301,12161,14331,16856,19789,23195,27139,31703,36978,43063,50075,58148
%N Coefficients of the A-Rogers mod 14 identity.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D 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.
%H G. C. Greubel, <a href="/A105780/b105780.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, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, ...]. - _Michael Somos_, Sep 21 2005
%F 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
%F 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
%F Number of partitions of n into parts all not == 0, 6, 8 (mod 14). - _Michael Somos_, Nov 21 2015
%F 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
%e 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 + ...
%e G.f. = 1/q + q^167 + 2*q^335 + 3*q^503 + 5*q^671 + 7*q^839 + 10*q^1007 + 14*q^1175 + ...
%t 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 *)
%t 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 *)
%o (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 */
%Y Cf. A105781, A105782.
%K nonn
%O 0,3
%A _Eric W. Weisstein_, Apr 19 2005
|
