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 A051136 #37 Jul 02 2025 16:01:58
%S A051136 1,4,9,20,42,80,147,260,445,744,1215,1944,3059,4740,7239,10920,16286,
%T A051136 24028,35110,50844,73010,104028,147144,206700,288501,400232,552037,
%U A051136 757288,1033495,1403508,1897088,2552812,3420527,4564500,6067265
%N A051136 Number of 2-colored generalized Frobenius partitions.
%C A051136 Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).
%D A051136 G. E. Andrews, "Generalized Frobenius Partitions," AMS Memoir 301, 1984 (sequence is denoted c\phi_2(n)).
%D A051136 G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 67, Eq. (7.20). MR0858826 (88b:11063)
%H A051136 G. C. Greubel, <a href="/A051136/b051136.txt">Table of n, a(n) for n = 0..1000</a>
%H A051136 Brian Drake, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953.
%H A051136 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A051136 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A051136 Expansion of phi(q) / f(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions.
%F A051136 Expansion of q^(1/12) * eta(q^2)^5 / (eta(q)^4 * eta(q^4)^2) in powers of q. - _Michael Somos_, Apr 25 2003
%F A051136 Euler transform of period 4 sequence [4, -1, 4, 1, ...]. - _Michael Somos_, Apr 25 2003
%F A051136 G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 24^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A137828.
%F A051136 G.f.: Product_{k>0} (1 -x^(4*k-2)) / ((1 - x^(2*k-1))^4 * (1 - x^(4*k))). [Andrews, Memoir, p. 13, equation (5.17)]
%F A051136 G.f.: Product_{k>0} (1 + x^k)^3 / ((1 - x^k) * (1 + x^(2*k))^2). - _Michael Somos_, Feb 12 2008
%F A051136 a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*sqrt(3)*n). - _Vaclav Kotesovec_, Aug 31 2015
%e A051136 1 + 4*x + 9*x^2 + 20*x^3 + 42*x^4 + 80*x^5 + 147*x^6 + 260*x^7 + ...
%e A051136 1/q + 4*q^11 + 9*q^23 + 20*q^35 + 42*q^47 + 80*q^59 + 147*q^71 + ...
%t A051136 nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1)) / ((1 - x^(2*k-1))^3 * (1 - x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 31 2015 *)
%t A051136 QP = QPochhammer; s = QP[q^2]^5 / QP[q]^4 / QP[q^4]^2 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 09 2015, adapted from PARI *)
%o A051136 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / eta(x + A)^4 / eta(x^4 + A)^2, n))} /* _Michael Somos_, Feb 12 2008 */
%Y A051136 Cf. A000041, A053762, A247663, A247664.
%K A051136 easy,nonn
%O A051136 0,2
%A A051136 _James Sellers_