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 A293204 #25 Oct 05 2017 11:39:27
%S A293204 1,1,3,2,6,7,12,13,22,26,42,46,73,80,116,139,194,226,306,358,482,558,
%T A293204 735,856,1108,1300,1657,1926,2426,2834,3530,4110,5082,5898,7234,8409,
%U A293204 10216,11860,14304,16568,19891,22990,27470,31670,37630,43382,51274,58982,69450
%N A293204 G.f.: Product_{m>0} (1+x^m+2!*x^(2*m)).
%H A293204 Alois P. Heinz, <a href="/A293204/b293204.txt">Table of n, a(n) for n = 0..5000</a> (first 501 terms from Seiichi Manyama)
%F A293204 a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4 * sqrt(Pi) * n^(3/4)), where c = Pi^2/3 - arctan(sqrt(7))^2 + log(2)^2/4 + polylog(2, -1/4 - I*sqrt(7)/4) + polylog(2, -1/4 + I*sqrt(7)/4) = 1.323865936864425754643630663383779192757247984691212163137... - _Vaclav Kotesovec_, Oct 02 2017
%F A293204 Equivalently, c = -polylog(2, -1/2 + I*sqrt(7)/2) - polylog(2, -1/2 - I*sqrt(7)/2). - _Vaclav Kotesovec_, Oct 05 2017
%e A293204 Let's consider the partitions of n where no positive integer appears more than twice. (See A000726)
%e A293204 For n = 5,
%e A293204 partition | |
%e A293204 --------------------------------------------------------------
%e A293204 5 -> one 5 -> 1! (= 1)
%e A293204 = 4 + 1 -> one 4 and one 1 -> 1!*1! (= 1)
%e A293204 = 3 + 2 -> one 3 and one 2 -> 1!*1! (= 1)
%e A293204 = 3 + 1 + 1 -> one 3 and two 1 -> 1!*2! (= 2)
%e A293204 = 2 + 2 + 1 -> two 2 and one 1 -> 2!*1! (= 2)
%e A293204 --------------------------------------------------------------
%e A293204 a(5) = 7.
%e A293204 For n = 6,
%e A293204 partition | |
%e A293204 --------------------------------------------------------------
%e A293204 6 -> one 6 -> 1! (= 1)
%e A293204 = 5 + 1 -> one 5 and one 1 -> 1!*1! (= 1)
%e A293204 = 4 + 2 -> one 4 and one 2 -> 1!*1! (= 1)
%e A293204 = 4 + 1 + 1 -> one 4 and two 1 -> 1!*2! (= 2)
%e A293204 = 3 + 3 -> two 3 -> 2! (= 2)
%e A293204 = 3 + 2 + 1 -> one 3, one 2 and one 1 -> 1!*1!*1! (= 1)
%e A293204 = 2 + 2 + 1 + 1 -> two 2 and two 1 -> 2!*2! (= 4)
%e A293204 --------------------------------------------------------------
%e A293204 a(6) = 12.
%p A293204 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A293204 add(b(n-i*j, i-1)*j!, j=0..min(2, n/i))))
%p A293204 end:
%p A293204 a:= n-> b(n$2):
%p A293204 seq(a(n), n=0..50); # _Alois P. Heinz_, Oct 02 2017
%t A293204 nmax = 100; CoefficientList[Series[Product[1 + x^k + 2*x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 02 2017 *)
%Y A293204 Column k=2 of A293202.
%Y A293204 Cf. A000726, A263401, A293138, A293182.
%Y A293204 Cf. A293072.
%K A293204 nonn
%O A293204 0,3
%A A293204 _Seiichi Manyama_, Oct 02 2017