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Strehl's sequence "C_n^(3)".
3

%I #27 Sep 09 2015 23:51:09

%S 1,4,68,1732,51076,1657904,57793316,2117525792,80483121028,

%T 3147565679824,125937573689968,5133632426499152,212530848994367524,

%U 8914634034287235856,378138515326996979168,16196097181014298854032

%N Strehl's sequence "C_n^(3)".

%D Volker Strehl, Binomial identities - combinatorial and algorithmic aspects. Trends in discrete mathematics. Discrete Math. 136 (1994), no. 1-3, 309-346.

%H Reinhard Zumkeller, <a href="/A000658/b000658.txt">Table of n, a(n) for n = 0..100</a>

%H Vaclav Kotesovec, <a href="/A000658/a000658.txt">Recurrence (of order 6)</a>

%F Sum binomial(n, k)^2 * binomial(2k, k)^2 * binomial(2k, n-k); k=0..n.

%F a(n) ~ 7^(2*n+5/2) / (20 * sqrt(15) * Pi^2 * n^2). - _Vaclav Kotesovec_, Mar 09 2014

%p A000658:=n->add(binomial(n,k)^2*binomial(2*k,k)^2*binomial(2*k,n-k), k=0..n): seq(A000658(n), n=0..15); # _Wesley Ivan Hurt_, Sep 19 2014

%t Table[Sum[Binomial[n,k]^2 Binomial[2k,k]^2 Binomial[2k,n-k],{k,0,n}], {n,0,25}] (* _Harvey P. Dale_, Oct 19 2011 *)

%o (PARI) a(n)=sum(k=1,n, binomial(n, k)^2 * binomial(2k, k)^2 * binomial(2k, n-k)) \\ _Charles R Greathouse IV_, Sep 19 2014

%o (Haskell)

%o a000658 n = sum $ map c3 [0..n] where

%o c3 k = (a007318' n k)^2 * (a007318' (2*k) k)^2 * a007318' (2*k) (n-k)

%o -- _Reinhard Zumkeller_, May 20 2015

%Y Third row of array A094424.

%Y Cf. A007318.

%K nonn,easy,nice

%O 0,2

%A _Don Knuth_