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 A112478 #43 Oct 08 2023 10:50:13
%S A112478 1,2,-2,6,-22,90,-394,1806,-8558,41586,-206098,1037718,-5293446,
%T A112478 27297738,-142078746,745387038,-3937603038,20927156706,-111818026018,
%U A112478 600318853926,-3236724317174,17518619320890,-95149655201962,518431875418926,-2832923350929742,15521467648875090
%N A112478 Expansion of (1 + x + sqrt(1 + 6*x + x^2))/2.
%C A112478 This is the A-sequence for the Delannoy triangle A008288. See the W. Lang link under A006232 for Sheffer a- and z-sequences where also Riordan A- and Z-sequences are explained. O.g.f. A(y) = y/Finv(y) = 2*y/(-(1 + y) + sqrt(y^2 + 6*y + 1)) = ((1 + y) + sqrt(1 + 6*y + y^2))/2 with Finv the inverse function of F(x) = x*(1 + x)/(1 - x). The o.g.f. of the Z-sequence is 1.
%H A112478 Vincenzo Librandi, <a href="/A112478/b112478.txt">Table of n, a(n) for n = 0..200</a>
%F A112478 G.f.: (1 + x + sqrt(1 + 6*x + x^2))/2. - _Sergei N. Gladkovskii_, Jan 04 2012
%F A112478 G.F.: G(0) where G(k)= 1 + x + x/G(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Jan 04 2012
%F A112478 D-finite with recurrence: n*a(n) + 3*(2*n-3)*a(n-1) + (n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 24 2012
%F A112478 a(n) ~ (-1)^(n+1) * sqrt(3*sqrt(2) - 4) * (3 + 2*sqrt(2))^n / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 12 2014
%F A112478 0 = a(n)*(a(n+1) + 15*a(n+2) + 4*a(n+3)) + a(n+1)*(-3*a(n+1) + 34*a(n+3) + 15*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all integer n > 0. - _Michael Somos_, Jul 07 2020
%F A112478 From _Seiichi Manyama_, Oct 08 2023: (Start)
%F A112478 G.f. satisfies A(x) = 1 + x + x/A(x).
%F A112478 a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(n+k-2,n-k)/(2*k-1). (End)
%e A112478 G.f. = 1 + 2*x - 2*x^2 + 6*x^3 - 22*x^4 + 90*x^5 - 394*x^6 + 1806*x^7 + ...
%t A112478 CoefficientList[Series[(1+x+Sqrt[1+6*x+x^2])/2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%o A112478 (PARI) {a(n) = polcoeff((1 + x + sqrt(1 + 6*x + x^2 + x*O(x^n)))/2, n)}; /* _Michael Somos_, Jul 07 2020 */
%Y A112478 A minor variation of A006318. See A085403 for yet another version.
%Y A112478 Row sums of number triangle A112477.
%Y A112478 Cf. A364393, A364407, A364408, A364409, A366266, A366267, A366268.
%Y A112478 Cf. A366325.
%K A112478 easy,sign
%O A112478 0,2
%A A112478 _Paul Barry_, Sep 07 2005