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 A107231 #25 Sep 04 2024 10:56:32
%S A107231 1,3,12,30,90,210,560,1260,3150,6930,16632,36036,84084,180180,411840,
%T A107231 875160,1969110,4157010,9237800,19399380,42678636,89237148,194699232,
%U A107231 405623400,878850700,1825305300,3931426800,8143669800,17450721000
%N A107231 a(n) = C(n+2,2)*C(n,floor(n/2)).
%C A107231 Third column of A107230. Related to the generalized pentagonal numbers A001318. The sequence 0,0,1,3,12,... is an inverse Chebyshev transform of 0,0,1,3,8,... (see A034828). This transform maps a g.f. g(x) to (1/sqrt(1-4x^2))g(c(x^2)). Thus A001318, as first differences of A034828, can be expressed in terms of A107231.
%H A107231 G. C. Greubel, <a href="/A107231/b107231.txt">Table of n, a(n) for n = 0..1000</a>
%F A107231 G.f.: (1+x)*(1-sqrt(1-4*x^2))^3*(sqrt(1-4*x^2)-4*x^2+1)^2/(8*x^4*(1-4*x^2)^(5/2)*(sqrt(1-4*x^2)+2*x-1)^2).
%F A107231 a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n+2, k)*A034828(n+2-2*k). [corrected by _Jason Yuen_, Sep 02 2024]
%F A107231 Conjecture: n*a(n) +(n-4)*a(n-1) +2*(-2*n-5)*a(n-2) -4*n*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012
%F A107231 G.f.: (1+x)/((1+2*x)^(3/2)*(1-2*x)^(5/2)). - _Vladimir Reshetnikov_, Aug 01 2018
%F A107231 Sum_{n>=0} 1/a(n) = Pi^2/9 - 2*Pi/sqrt(3) + 4. - _Amiram Eldar_, Sep 03 2024
%t A107231 Table[Binomial[n + 2, 2]*Binomial[n, Floor[n/2]], {n,0,50}] (* _G. C. Greubel_, Jun 13 2017 *)
%o A107231 (PARI) for(n=0,50, print1(binomial(n+2,2)*binomial(n,n\2), ", ")) \\ _G. C. Greubel_, Jun 13 2017
%Y A107231 Cf. A000217, A001405.
%Y A107231 Cf. A001318, A034828, A107230.
%K A107231 easy,nonn
%O A107231 0,2
%A A107231 _Paul Barry_, May 13 2005