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A103882 a(n) = Sum_{i=0..n} C(n+1,i)*C(n-1,i-1)*C(2n-i,n). 4

%I

%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 R. H. Hardin and 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 Supercongruences: 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) = hypergeom([-n, -n, n], [1, 1], 1). - _Peter Luschny_, Jan 19 2020

%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 *)

%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

%Y Equals A103881(n, n). Cf. A005258, A177316.

%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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Last modified August 3 20:08 EDT 2020. Contains 336201 sequences. (Running on oeis4.)