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a(n) = T(2*n, n), T given by A027907.
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%I #49 May 01 2024 09:01:20

%S 1,2,10,50,266,1452,8074,45474,258570,1481108,8533660,49402850,

%T 287134346,1674425300,9792273690,57407789550,337281021450,

%U 1985342102964,11706001102180,69124774458092,408737856117916,2419833655003752,14341910428953018,85087759173024870

%N a(n) = T(2*n, n), T given by A027907.

%C Central terms of the triangle in A111808. - _Reinhard Zumkeller_, Aug 17 2005

%C Number of paths of semilength n starting at (0,0) and ending on the X-axis using steps (1,1), (1,-1) and (1,3). - _David Scambler_, Jun 21 2013

%H Seiichi Manyama, <a href="/A027908/b027908.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: -(g^2+g+1)/(3*g^2+g-1) where g = x times the g.f. of A143927. - _Mark van Hoeij_, Nov 16 2011

%F a(n) = GegenbauerC(n, -2*n, -1/2). - _Peter Luschny_, May 09 2016

%F From _Peter Bala_, Jan 26 2020: (Start)

%F a(n) = [x^(2*n)](1 + x^2 + x^4)^(2*n).

%F a(n) = Sum_{k = 0..floor(n/2)} C(2*n, n-k)*C(n-k, k).

%F a(n) = C(2*n,n) * hypergeom([-n/2, (1 - n)/2], [n + 1], 4)

%F Conjectural: a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for all primes p >= 5 and positive integers n and k. (End)

%F From _Peter Bala_, Aug 03 2023: (Start)

%F P-recursive: 3*n*(13*n - 17)*(3*n - 1)*(3*n - 2)*a(n) = 2*(2*n - 1)*(455*n^3 - 1050*n^2 + 691*n - 120)*a(n-1) + 36*(n - 1)*(13*n - 4)*(2*n - 1)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 2.

%F exp(Sum_{n >= 0} a(n)*x^n/n) = 1 + 2*x + 7*x^2 + 28*x^3 + 123*x^4 + ... is the g.f. of A143927.

%F a(n) = 2*A344396(n-1) for n >= 1. (End)

%p ogf := series( RootOf( (144*x^2+140*x-27)*g^4+(18-12*x)*g^2+8*g+1, g), x=0, 20); # _Mark van Hoeij_, Nov 16 2011

%p a := n -> simplify(GegenbauerC(n, -2*n, -1/2)):

%p seq(a(n), n=0..23); # _Peter Luschny_, May 09 2016

%t Table[Binomial[4 n, n] Hypergeometric2F1[-3 n, -n, 1/2 - 2 n, 1/4], {n, 0, 20}] (* or *) Table[GegenbauerC[3 n, -2 n, -1/2] + KroneckerDelta[n], {n, 0, 20}] (* _Vladimir Reshetnikov_, May 07 2016 *)

%o (Maxima) makelist(ultraspherical(n,-2*n,-1/2),n,0,12); /* _Emanuele Munarini_, Oct 18 2016 */

%Y Cf. A027907, A027913, A143927, A344396, A370159, A370160.

%K nonn

%O 0,2

%A _Clark Kimberling_