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%I #64 Apr 04 2024 07:53:54
%S 1,1,4,9,26,70,197,553,1570,4476,12827,36894,106471,308113,893804,
%T 2598313,7567466,22076404,64498427,188689684,552675365,1620567763,
%U 4756614062,13974168190,41088418151,120906613075,356035078102
%N Expansion of 2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2))).
%C Convolution of A002426 and A005043. Diagonal sums of A094531.
%C Hankel transform is A164611. - _Paul Barry_, Aug 17 2009
%C _David Scambler_ observed that [1,0,a(n-2)] for n>=2 count the Dyck paths of semilength n such that the number of peaks equals the number of hills plus the number of returns. - _Peter Luschny_, Oct 22 2012
%C Conjectural congruences (working with an offset of 1): a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for prime p >= 5 and positive integers n and k. - _Peter Bala_, Mar 15 2020
%H G. C. Greubel, <a href="/A113682/b113682.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%F a(n) = Sum_{k=0..floor(n/2)} ( Sum_{i=0..n-k} C(n-2k-i, i)*C(n-k, k+i) ).
%F a(n) = Sum_{k=0..n} A002426(k)*A005043(n-k).
%F a(n) = Sum_{k=0..n} C(n+1,k+1)*C(k,n-k). - _Paul Barry_, Aug 21 2007
%F a(n) = (A002426(n+1) + (-1)^n)/2. - _Paul Barry_, Aug 17 2009
%F G.f.: d/dx log(1/(1-x*A005043(x))). - _Vladimir Kruchinin_, Apr 18 2011
%F D-finite with recurrence: (n+1)*a(n) +(-n-1)*a(n-1) +(-5*n+1)*a(n-2) +3*(-n+1)*a(n-3)=0. - _R. J. Mathar_, Nov 26 2012
%F Recurrence: (n+4)*a(n+3)-(n+4)*a(n+2)-(5*n+14)*a(n+1)-3*(n+2)*a(n)=0. Remark: this recurrence can be obtained using the identity a(n) = (t(n+1)+(-1)^n)/2 and the recurrence of the central trinomial coefficients t(n) = A002426(n). So, the above P-finite recurrences are true. - _Emanuele Munarini_, Dec 20 2016
%F a(n) = (-1)^(n+1) * (hypergeom([1/2, -n-1], [1], 4) - 1)/2. - _Vladimir Reshetnikov_, Apr 25 2016
%F a(n) = (-1)^n + A246437(n+1). - _Vladimir Reshetnikov_, Apr 25 2016
%t ex[x_]:=Module[{sx=Sqrt[1-2x-3x^2]},2/(sx (1+x+sx))]; CoefficientList[ Series[ ex[x],{x,0,40}],x] (* _Harvey P. Dale_, May 28 2012 *)
%t Flatten[{1, Table[Coefficient[Sum[(1 + x + x^2)^k, {k, 0, n}], x^n], {n, 1, 30}]}] (* _Vaclav Kotesovec_, Jan 08 2016 *)
%o (Maxima) makelist((ultraspherical(n+1,-n-1,-1/2)+(-1)^n)/2,n,0,12); /* _Emanuele Munarini_, Dec 20 2016 */
%o (PARI) x='x+O('x^50); Vec(2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2)))) \\ _G. C. Greubel_, Feb 28 2017
%o (Magma) [(Evaluate(GegenbauerPolynomial(n+1, -n-1), -1/2) + (-1)^n)/2: n in [0..40]]; // _G. C. Greubel_, Apr 04 2024
%o (SageMath) [(gegenbauer(n+1,-n-1,-1/2) +(-1)^n)/2 for n in range(41)] # _G. C. Greubel_, Apr 04 2024
%Y Cf. A002426, A005043, A094531, A217539, A217540, A246437.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Nov 04 2005
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