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 A098151 #80 Jul 19 2025 14:53:32
%S A098151 1,2,4,6,10,16,24,36,52,74,104,144,198,268,360,480,634,832,1084,1404,
%T A098151 1808,2316,2952,3744,4728,5946,7448,9294,11556,14320,17688,21780,
%U A098151 26740,32736,39968,48672,59122,71644,86616,104484,125768,151072,181104,216684
%N A098151 Number of partitions of 2*n with no part divisible by 3 and all odd parts occurring with even multiplicities.
%C A098151 There are no partitions of 2n+1 in which all odd parts occur with even multiplicity. - _Michael Somos_, Apr 15 2012
%C A098151 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A098151 a(n) is also the number of Schur overpartitions of n, i.e., the number of overpartitions of n where adjacent parts differ by at least 3 if the smaller is overlined or divisible by 3 and adjacent parts differ by at least 6 if the smaller is overlined and divisible by 3. - _Jeremy Lovejoy_, Mar 23 2015
%C A098151 Let A(q) denote the g.f. of this sequence. Let m be a nonzero integer. The simple continued fraction expansions of the real numbers A(1/(2*m)) and A(1/(2*m+1)) may be predictable. For a given positive integer n, the sequence of the n-th partial denominators of the continued fractions are conjecturally polynomial or quasi-polynomial in m for m sufficiently large. An example is given below. Cf. A080054. - _Peter Bala_, Jun 09 2025
%H A098151 Vaclav Kotesovec, <a href="/A098151/b098151.txt">Table of n, a(n) for n = 0..2000</a>
%H A098151 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/315.pdf">4-Shadows in q-Series and the Kimberling Index</a>, Preprint, May 15, 2016.
%H A098151 Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011, 2004.
%H A098151 Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, <a href="https://arxiv.org/abs/2507.10965">Convolutive sequences, I: Through the lens of integer partition functions</a>, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 14-16.
%H A098151 Byungchan Kim and Eunmi Kim, <a href="https://doi.org/10.4134/BKMS.b230752">Partitions weighted by the number of two types of parts</a>, Bull. Korean Math. Soc. (2024) Vol. 61, No. 6, 1677-1684. See p. 1679.
%H A098151 Jeremy Lovejoy, <a href="http://lovejoy.perso.math.cnrs.fr/overschur3.pdf">A theorem on seven-colored overpartitions and its applications</a>, Int. J. Number Theory. 1 (2005) 215-224
%H A098151 Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.
%H A098151 Andrew Sills, <a href="http://home.dimacs.rutgers.edu/~asills/EMDC/SillsEMDC-Rev.pdf">Towards an Automation of the Circle Method</a>, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S24.
%H A098151 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A098151 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A098151 Expansion of phi(-q^3) / phi(-q) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Apr 15 2012
%F A098151 Expansion of f(q, q^2) / f(-q, -q^2) in powers of q where f(,) is the Ramanujan two-variable theta function. - _Michael Somos_, Apr 15 2012
%F A098151 Expansion of eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
%F A098151 G.f. = (Sum_{n = -oo..oo} (-1)^n*q^(3*n^2)) / (Sum_{n = -oo..oo} (-1)^n*q^(n^2)). - _N. J. A. Sloane_, Aug 09 2016
%F A098151 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u^2) * (u^2 + v^4) - 4 * u^2*v^4. - _Michael Somos_, Apr 15 2012
%F A098151 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3 * u*v^2 - 3 * u^2*v^3. - _Michael Somos_, Dec 04 2004
%F A098151 Euler transform of period 6 sequence [2, 1, 0, 1, 2, 0, ...]. - _Vladeta Jovovic_, Sep 24 2004
%F A098151 Taylor series of product_{k=1..inf}(1+x^k+x^(2*k))/(1-x^k+x^(2*k))= product_{k=1..inf}(1+x^k)(1-x^(3k))/((1-x^k)(1+x^(3k)))=Theta_4(0, x^3)/theta_4(0, x)
%F A098151 a(n) ~ Pi * BesselI(1, Pi*sqrt(2*n/3)) / (3*sqrt(2*n)) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)) * (1 - 3*sqrt(3)/(8*Pi*sqrt(2*n)) - 45/(256*Pi^2*n)). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 09 2017
%F A098151 Convolution of A000726 and A003105. - _R. J. Mathar_, Nov 17 2017
%F A098151 From _Peter Bala_, Jun 09 2025: (Start)
%F A098151 G.f.: A(q) = Sum_{n = -oo..oo} q^(n*(3*n+1)/2) / Sum_{n = -oo..oo} (-1)^n * q^(n*(3*n+1)/2).
%F A098151 Recurrences:
%F A098151 a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + + - - ... = f(n), where [0, 1, 2, 5, 7, 12, 15, ...] is the sequence of generalized pentagonal numbers A001318, a(n) is set equal to 0 for negative n and f(n) = 1 if n is a generalized pentagonal number, otherwise f(n) = 0 (see A080995). Compare with the recurrence for the partition function p(n) = A000041(n).
%F A098151 a(n) - 2*a(n-1) + 2*a(n-4) - 2*a(n-9) + 2*a(n-16) - 2*a(n-25) + - ... = g(n), where g(n) = 2*(-1)^k if n is of the form 3*(k^2), otherwise g(n) = 0. (End)
%e A098151 a(4)=10 because 8 = 4+4 = 4+2+2=2+2+2+2 = 2+2+2+1+1 = 2+2+1+1+1+1 = 4+2+1+1 = 4+1+1+1+1 = 2+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1.
%e A098151 G.f. = 1 + 2*q + 4*q^2 + 6*q^3 + 10*q^4 + 16*q^5 + 24*q^6 + 36*q^7 + 52*q^8 + ...
%e A098151 From _Peter Bala_, Jun 09 2025: (Start)
%e A098151 G.f.: A(q) = f(q, q^2) / f(-q, -q^2).
%e A098151 Simple continued fraction expansions of A(1/(2*m)):
%e A098151 m = 2 [1; 1 9 1 5 8 45 4 1 2 1 1 1 3 3 2 2 ...]
%e A098151 m = 3 [1; 2 13 1 14 12 133 8 1 1 7 2 1 2 2 1 1 ...]
%e A098151 m = 4 [1; 3 17 1 27 16 297 12 2 2 1 1 1 2 2 2 2 ...]
%e A098151 m = 5 [1; 4 21 1 44 20 561 16 2 1 7 3 3 2 2 25 8 ...]
%e A098151 m = 6 [1; 5 25 1 65 24 949 20 3 2 1 1 1 3 4 2 1 ...]
%e A098151 m = 7 [1; 6 29 1 90 28 1485 24 3 1 7 4 5 2 1 1 6 ...]
%e A098151 m = 8 [1; 7 33 1 119 32 2193 28 4 2 1 1 1 4 6 2 1 ...]
%e A098151 m = 9 [1; 8 37 1 152 36 3097 32 4 1 7 5 7 2 1 1 3 ...]
%e A098151 m = 10 [1; 9 41 1 189 40 4221 36 5 2 1 1 1 5 8 2 1 ...]
%e A098151 ...
%e A098151 The sequence of the 4th partial denominators [5, 14, 27, 44, ...] appears to be given by the polynomial (2*m + 1)*(m - 1) for m >= 2.
%e A098151 The sequence of the 6th partial denominators [45, 133, 297, 561, ...] appears to be given by the polynomial (2*m + 1)*(2*m^2 + 1) for m >= 2. (End)
%p A098151 series(product((1+x^k+x^(2*k))/(1-x^k+x^(2*k)),k=1..150),x=0,100);
%p A098151 # alternative program using expansion of f(q, q^2) / f(-q, -q^2):
%p A098151 with(gfun): series( add(x^(n*(3*n-1)/2),n = -8..8)/add((-1)^n*x^(n*(3*n-1)/2), n = -8..8), x, 100): seriestolist(%); # _Peter Bala_, Feb 05 2021
%t A098151 a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^3]^2 / (QPochhammer[ q]^2 QPochhammer[ q^6]), {q, 0, n}] (* _Michael Somos_, Oct 23 2013 *)
%t A098151 nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1)) * (1+x^(3*k-2)) / ((1-x^(3*k-1)) * (1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 31 2015 *)
%o A098151 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))} /* _Michael Somos_, Dec 04 2004 */
%Y A098151 Cf. A015128, A001318, A080054, A080995, A010815, A103260, A215594.
%K A098151 nonn,easy
%O A098151 0,2
%A A098151 _Noureddine Chair_, Aug 29 2004