A092833 - OEIS

%I #26 Mar 12 2021 22:24:42

%S 1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,32,38,46,54,64,76,89,105,123,

%T 143,167,194,225,260,301,346,398,458,524,600,686,782,891,1014,1151,

%U 1306,1480,1674,1892,2137,2409,2713,3053,3431,3852,4322,4842,5421,6064,6776

%N Expansion of q / (chi(-q) * chi(-q^23)) in powers of q where chi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q)(A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A092833/b092833.txt">Table of n, a(n) for n = 1..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>

%F G.f.: x * (Product_{k>0} (1 + x^k) * (1 + x^(23*k))).

%F Expansion of eta(q^2) * eta(q^46) / (eta(q) * eta(q^23)) in powers of q.

%F Euler transform of period 46 sequence with g.f. x / (1 - x^2) + x^23 / (1 - x^46).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 2 * u*v * (1 + v).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = (1/2)  g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132322.

%F Convolution inverse of A132322.

%F a(n) = A112216(2*n). - _Michael Somos_, Aug 11 2015

%F a(n) ~ exp(2*Pi*sqrt(2*n/23)) / (2^(7/4) * 23^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015

%e G.f. = q + q^2 + q^3 + 2*q^4 + 2*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + 8*q^10 + ...

%t a[n_] := Coefficient[ Series[ x*Product[(1 + x^k)*(1 + x^(23*k)), {k, 1, n}], {x, 0, n}], x, n]; Table[a[n], {n, 1, 56}] (* _Jean-François Alcover_, Jan 28 2013, from 1st formula *)

%t a[ n_] := SeriesCoefficient[ q Product[ (1 + q^k) (1 + q^(23 k)), {k, n}], {q, 0, n}]; (* _Michael Somos_, Aug 11 2015 *)

%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q, q]  QPochhammer[ -q^23, q^23]), {q, 0, n}]; (* _Michael Somos_, Aug 11 2015 *)

%o (PARI) {a(n) = my(A, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = A + A^2 + sqrt(A + (A + A^2)^2)); polcoeff(A, n))};

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^46 + A) / eta(x + A) / eta(x^23 + A), n))};

%Y Cf. A112216, A132322.

%K nonn

%O 1,4

%A _Michael Somos_, Mar 06 2004