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 A105781 Coefficients of the B-Rogers mod 14 identity. 3
 1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 30, 40, 53, 68, 88, 113, 143, 181, 228, 284, 354, 439, 541, 665, 815, 993, 1208, 1465, 1769, 2132, 2563, 3070, 3671, 4379, 5209, 6185, 7329, 8663, 10223, 12041, 14153, 16609, 19459, 22755, 26571, 30979, 36059, 41915, 48654 (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). 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, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, ...]. - Michael Somos, Sep 21 2005 G.f.: Product_{k>0} (1 - x^(14*k)) * (1 - x^(14*k - 4)) * (1 - x^(14*k - 10)) / (1 - x^k) = Sum_{k>=0} x^(k^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^4, -x^10) / 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, 4, 10 (mod 14). - Michael Somos, Nov 21 2015 a(n) ~ 11^(1/4) * cos(3*Pi/14) * 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 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 17*x^8 + 23*x^29 + ... G.f. = q^47 + q^215 + 2*q^383 + 3*q^551 + 4*q^719 + 6*q^887 + 9*q^1055 + 12*q^1223 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ x^4, x^14] QPochhammer[ x^10, 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, 0, 1, 1, 1, 1, 1, 0, 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, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1][k%14 + 1] * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Sep 21 2005 */ CROSSREFS Cf. A105780, A105782. Sequence in context: A248475 A035952 A335754 * A035958 A035965 A035973 Adjacent sequences:  A105778 A105779 A105780 * A105782 A105783 A105784 KEYWORD nonn AUTHOR Eric W. Weisstein, Apr 19 2005 STATUS approved

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Last modified August 9 18:32 EDT 2020. Contains 336326 sequences. (Running on oeis4.)