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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 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.)