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 A091147 #54 Nov 03 2024 02:14:27
%S A091147 1,1,5,13,57,201,861,3445,14897,63313,278389,1223069,5465065,24513945,
%T A091147 111037005,505298565,2314343265,10645982625,49202944485,228253816365,
%U A091147 1062783893145,4964167491945,23256852644925,109249893866133,514494575459217,2428488338526961
%N A091147 Expansion of (1-x-sqrt(1-2x-15x^2))/(8x^2).
%C A091147 a(n) = A014433(n+1)/4.
%C A091147 Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in 4 colors (i.e. Motzkin paths with the up steps in 4 colors). Series reversion of x/(1+x+4x^2). - _Paul Barry_, May 16 2005
%H A091147 Vincenzo Librandi, <a href="/A091147/b091147.txt">Table of n, a(n) for n = 0..200</a>
%F A091147 G.f.: 2/(1-x+sqrt(1-2x-15x^2)).
%F A091147 G.f.: exp( Sum_{n>=1} A084605(n) * x^n/n ). - _Paul D. Hanna_, Dec 08 2018
%F A091147 a(n) = sum{k=0..n, binomial(n, k)4^(k/2)C(k/2)(1+(-1)^k)/2}, C(n)=A000108(n).
%F A091147 a(n) = sum{k=0..n, C(n, 2k)C(k)4^k}. - _Paul Barry_, May 16 2005
%F A091147 a(n) = integral(x=-2..2, (2*x+1)^n*sqrt((2-x)*(2+x)))/(2*Pi). [_Peter Luschny_, Sep 11 2011]
%F A091147 a(n) = (2^n/(n+1))*[x^n] (1+x/2+x^2)^(n+1). [_Emanuele Munarini_, Apr 27 2012]
%F A091147 E.g.f.: a(n) = n! * [x^n] exp(x)*BesselI(1, 4*x)/(2*x). -_Peter Luschny_, Aug 25 2012
%F A091147 D-finite with recurrence: (n+2)*a(n) -(2*n+1)*a(n-1) +15*(1-n)*a(n-2)=0. - _R. J. Mathar_, Sep 26 2012, [corrected by _Vaclav Kotesovec_, Sep 29 2012]
%F A091147 a(n) ~ 5/8*sqrt(10)/(n^(3/2)*sqrt(Pi))*5^n. - _Vaclav Kotesovec_, Sep 29 2012
%F A091147 a(n) = hypergeom([-n/2, (1-n)/2], [2], 16). - _Peter Luschny_, May 28 2014
%F A091147 a(n) = 2^n*GegenbauerC(n,-n-1, -1/4)/(n+1). - _Peter Luschny_, May 08 2016
%F A091147 G.f.: 1/(1 - x - 4*x^2/(1 - x - 4*x^2/(1 - x - 4*x^2/(1 - x - 4*x^2/(1 - ....))))), a continued fraction. - _Ilya Gutkovskiy_, May 26 2017
%p A091147 a := n -> simplify(2^n*GegenbauerC(n, -n-1, -1/4)/(n+1)):
%p A091147 seq(a(n), n=0..25); # _Peter Luschny_, May 08 2016
%t A091147 a[0] = 1; a[1] = 1; a[n_] := ((2*n + 1)*a[n - 1] - 15*(1 - n)*a[n - 2])/(n + 2); Table[a[n], {n, 0, 50}] (* _T. D. Noe_, Oct 02 2012 *)
%t A091147 CoefficientList[Series[(1 - x - Sqrt[1 - 2 x - 15 x^2]) / (8 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, May 10 2013 *)
%t A091147 a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 2, 16];
%t A091147 Table[a[n], {n, 0, 25}] (* _Peter Luschny_, Mar 18 2018 *)
%o A091147 (Maxima) a(n):=2^n*coeff(expand((1+x/2+x^2)^(n+1)),x^n)/(n+1);
%o A091147 makelist(a(n),n,0,30); /* Emanuele Munarini, Apr 27 2012 */
%o A091147 (PARI) my(x='x+O('x^66)); Vec((1-x-sqrt(1-2*x-15*x^2))/(8*x^2)) \\ _Joerg Arndt_, May 11 2013
%Y A091147 Cf. A217275, A084605.
%K A091147 easy,nonn
%O A091147 0,3
%A A091147 _Paul Barry_, Dec 22 2003