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 A105780 Coefficients of the A-Rogers mod 14 identity. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified May 26 17:16 EDT 2020. Contains 334630 sequences. (Running on oeis4.)