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 A036026 #22 Feb 16 2025 08:32:37
%S A036026 1,1,1,2,3,3,4,6,8,10,12,16,21,25,30,38,48,57,68,84,102,121,143,172,
%T A036026 207,243,284,338,400,465,542,636,744,862,996,1158,1344,1546,1776,2050,
%U A036026 2361,2701,3088,3540,4050,4613,5248,5980,6808,7719,8742,9916,11232
%N A036026 Number of partitions of n into parts not of forms 4*k+2, 20*k, 20*k+5 or 20*k+15.
%C A036026 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A036026 Case k=5,i=3 of Gordon/Goellnitz/Andrews Theorem.
%C A036026 Number of partitions in which no odd part is repeated, with at most 2 parts of size less than or equal to 2 and where differences between parts at distance 4 are greater than 1 when the smaller part is odd and greater than 2 when the smaller part is even.
%D A036026 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
%H A036026 G. C. Greubel, <a href="/A036026/b036026.txt">Table of n, a(n) for n = 0..1000</a>
%H A036026 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A036026 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A036026 Expansion of psi(-x^5) / psi(-x) in powers of x where psi() is a Ramanujan theta function. - _Michael Somos_, Oct 27 2008
%F A036026 Expansion of q^(-1/2) * eta(q^2) * eta(q^5) * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10)) in powers of q. - _Michael Somos_, Oct 27 2008
%F A036026 Euler transform of period 20 sequence [ 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, ...]. - _Michael Somos_, Oct 27 2008
%F A036026 a(2*n) = A145722(n). a(2*n + 1) = A101277(n). Convolution inverse of A145708.
%F A036026 a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(5/4) * 5^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, May 10 2018
%e A036026 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + ...
%e A036026 q + q^3 + q^5 + 2*q^7 + 3*q^9 + 3*q^11 + 4*q^13 + 6*q^15 + 8*q^17 + ...
%t A036026 a[n_] := SeriesCoefficient[EllipticTheta[2, 0, I*q^(5/2)]/( q^(1/2) * EllipticTheta[2, 0, I*q^(1/2)]), {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Apr 16 2017 *)
%t A036026 nmax = 60; CoefficientList[Series[Product[(1 - x^(4*k - 2))*(1 - x^(20*k))*(1 - x^(20*k+5-20))*(1 - x^(20*k- 5))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, May 10 2018 *)
%o A036026 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))} /* _Michael Somos_, Oct 27 2008 */
%Y A036026 Cf. A101277, A145708, A145722.
%K A036026 nonn,easy
%O A036026 0,4
%A A036026 _Olivier Gérard_