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%I #91 Oct 27 2023 19:51:42
%S 1,2,12,92,780,7002,65226,623576,6077196,60110030,601585512,
%T 6078578508,61908797418,634756203018,6545498596110,67830161708592,
%U 705951252118284,7375213677918294,77310179609631564,812839595630249540,8569327862277434280,90562666977432643862
%N a(n) = Sum_{i=0..n} C(n+1,i)*C(n-1,i-1)*C(2n-i,n).
%C Number of permutations of n copies of 1..3 with all adjacent differences <= 1 in absolute value. - _R. H. Hardin_, May 06 2010 [Cf. A177316. - _Peter Bala_, Jan 14 2020]
%H Alois P. Heinz, <a href="/A103882/b103882.txt">Table of n, a(n) for n = 0..950</a> (terms n=1..94 from R. H. Hardin)
%H A. Straub, <a href="https://arxiv.org/abs/1401.0854">Multivariate Apéry numbers and supercongruences of rational functions</a>, arXiv:1401.0854 [math.NT] (2014).
%F a(n) = (A005258(n-1) + 3*A005258(n))/5 (Apéry numbers). - _Mark van Hoeij_, Jul 13 2010
%F n^2*(n-1)*(5*n-8)*a(n) = (n-1)*(55*n^3-143*n^2+102*n-24)*a(n-1) + n*(n-2)^2*(5*n-3)*a(n-2). - _Alois P. Heinz_, Jun 29 2015
%F a(n) ~ phi^(5*n + 3/2) / (2*Pi*5^(1/4)*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jul 21 2019
%F From _Peter Bala_, Jan 14 2020: (Start)
%F a(n) = Sum_{k = 0..n} C(n,k)^2*C(n+k-1,k). Cf. A005258.
%F For any prime p >= 5, a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all positive integers n and k (follows from known supercongruences satisfied by the Apéry numbers A005258 - see Straub, Example 3.4). (End)
%F a(n) = hypergeometric([-n, -n, n], [1, 1], 1). - _Peter Luschny_, Jan 19 2020
%F From _Peter Bala_, Dec 19 2020: (Start)
%F a(n) = Sum_{k = 1..n} C(n,k)*C(n+k,k)*C(n-1,k-1) for n >= 1.
%F a(n) = [x^n] P(n, (1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Cf. A156554. (End)
%F a(n) = Sum_{k = 0..n} binomial(2*n-k-1,n-k)* binomial(n,k)^2. Cf. A108628. - _Peter Bala_, Mar 24 2022
%F From _Peter Bala_, Apr 15 2022: (Start)
%F a(-n) = (-1)^n*A352654(n).
%F a(n) = [x^n*y^n*z^(n-1)] 1/(1 - x - y - z + x*z + y*z - x*y*z) for n >= 1.
%F a(n) = B(n,n,n-1) in the notation of Straub, see equation 24.
%F a(n) = [x^n*y^n*z^(n-1)] (x + y + z)^n*(x + y)^n*(y + z)^(n-1) for n >= 1. (End)
%F D-finite with recurrence 9*n^2*a(n) -3*(31*n^2-27*n+6)*a(n-1) -2*(37*n^2-138*n+108)*a(n-2) -(n-3)*(17*n-56)*a(n-3) -(n-4)^2*a(n-4) = 0. - _R. J. Mathar_, Aug 01 2022
%F a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n-1, n-k)* binomial(n+k, k)* binomial(n+k-1, k). - _Peter Bala_, Aug 13 2023
%p a:= proc(n) option remember; `if`(n<2, n+1,
%p ((n-1)*(55*n^3-143*n^2+102*n-24)*a(n-1)+
%p n*(5*n-3)*(n-2)^2*a(n-2))/((n-1)*(5*n-8)*n^2))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jun 29 2015
%p # Alternative:
%p a := n -> hypergeom([-n, -n, n], [1, 1], 1):
%p seq(simplify(a(n)), n=0..21); # _Peter Luschny_, Jan 19 2020
%t Drop[Table[Sum[Sum[Multinomial[r, g, n + 1 - r - g] Binomial[n - 1,n - r] Binomial[n - 1, n - g], {g, 1, n}], {r, 1, n}], {n, 0, 18}], 1] (* _Geoffrey Critzer_, Jun 29 2015 *)
%t Table[Sum[Binomial[n+1,k]Binomial[n-1,k-1]Binomial[2n-k,n],{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Jun 19 2021 *)
%o (Magma) [1] cat [&+[Binomial(n+1, i)*Binomial(n-1, i-1) * Binomial(2*n-i, n): i in [0..n]]:n in [1..21]]; // _Marius A. Burtea_, Jan 19 2020
%o (Magma) [&+[Binomial(n, k)^2*Binomial(n+k-1, k): k in [0..n]]:n in [0..21]]; // _Marius A. Burtea_, Jan 19 2020
%o (PARI) a(n) = polcoef(pollegendre(n, (1 + x)/(1 - x)) + O(x^(n+1)), n); \\ _Michel Marcus_, Dec 20 2020
%o (Python)
%o def A103882(n):
%o if n == 0: return 1
%o m, g = 1, 0
%o for k in range(n+1):
%o g += m*n//(n+k)
%o m *= (n+k+1)*(n-k)**2
%o m //= (k+1)**3
%o return g # _Chai Wah Wu_, Oct 04 2022
%o (SageMath)
%o def A103882(n): return hypergeometric([-n,-n,n], [1,1], 1).simplify()
%o [A103882(n) for n in range(31)] # _G. C. Greubel_, May 24 2023
%Y Cf. A005258, A156554, A177316, A352654.
%Y Equals A103881(n, n).
%Y Row n=3 of A331562.
%K nonn,easy
%O 0,2
%A _Ralf Stephan_, Feb 20 2005
%E a(0)=1 prepended by _Alois P. Heinz_, Jun 29 2015
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