This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A100823 #32 Feb 16 2025 08:32:55
%S A100823 1,2,4,7,12,19,30,46,69,101,146,208,293,408,563,768,1040,1397,1864,
%T A100823 2470,3254,4261,5550,7192,9277,11911,15229,19391,24597,31085,39150,
%U A100823 49142,61489,76702,95401,118324,146362,180573,222226,272826,334173,408394,498022
%N A100823 G.f.: Product_{k>0} (1+x^k)/((1-x^k)*(1+x^(3k))*(1+x^(5k))).
%C A100823 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A100823 Vaclav Kotesovec and Alois P. Heinz, <a href="/A100823/b100823.txt">Table of n, a(n) for n = 0..10000</a> (first 2001 terms from Vaclav Kotesovec)
%H A100823 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 17.
%H A100823 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A100823 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A100823 a(n) ~ exp(Pi*sqrt(37*n/5)/3) * sqrt(37) / (12*sqrt(5)*n). - _Vaclav Kotesovec_, Sep 01 2015
%F A100823 G.f.: (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) where E(k) = prod(n>=1, 1-q^k ). - _Joerg Arndt_, Sep 01 2015
%F A100823 Euler transform of period 30 sequence [ 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, ...]. - _Michael Somos_, Mar 07 2016
%F A100823 Expansion of chi(-x^3) * chi(-x^5) / phi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - _Michael Somos_, Mar 07 2016
%F A100823 a(n) - A035939(2*n + 1) = A122129(2*n + 1). - _Michael Somos_, Mar 07 2016
%e A100823 G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 46*x^7 + ...
%e A100823 G.f. = q^-1 + 2*q^2 + 4*q^5 + 7*q^8 + 12*q^11 + 19*q^14 + 30*q^17 + 46*q^20 + ...
%p A100823 series(product((1+x^k)/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))),k=1..100),x=0,100);
%t A100823 CoefficientList[ Series[ Product[(1 + x^k)/((1 - x^k)*(1 + x^(3k))*(1 + x^(5k))), {k, 100}], {x, 0, 45}], x] (* _Robert G. Wilson v_, Jan 12 2005 *)
%t A100823 nmax = 50; CoefficientList[Series[Product[(1+x^(5*k-1))*(1+x^(5*k-2))*(1+x^(5*k-3))*(1+x^(5*k-4)) / ((1-x^(6*k))*(1-x^(3*k-1))*(1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 01 2015 *)
%t A100823 a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] QPochhammer[ x^5, x^10] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* _Michael Somos_, Mar 07 2016 *)
%o A100823 (PARI) q='q+O('q^33); E(k)=eta(q^k);
%o A100823 Vec( (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) ) \\ _Joerg Arndt_, Sep 01 2015
%Y A100823 Cf. A035939, A098151, A102346, A122129.
%K A100823 nonn
%O A100823 0,2
%A A100823 _Noureddine Chair_, Jan 06 2005
%E A100823 More terms from _Robert G. Wilson v_, Jan 12 2005
%E A100823 Offset corrected by _Vaclav Kotesovec_, Sep 01 2015
%E A100823 a(14) = 563 <- 562 corrected by _Vaclav Kotesovec_, Sep 01 2015