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A027277 a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(2*n-k,k).

%I A027277 #32 Sep 08 2022 08:44:49
%S A027277 1,3,13,67,375,2189,13089,79479,487833,3018355,18792303,117589689,
%T A027277 738844719,4658460165,29458662005,186761788579,1186655988771,
%U A027277 7554520173441,48176764031385,307706150625855,1968040844127793,12602972755261195,80798365998084795,518536437750443773
%N A027277 a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(2*n-k,k).
%C A027277 Previous name was: a(n) = self-convolution of row n of array T given by A026568.
%H A027277 G. C. Greubel, <a href="/A027277/b027277.txt">Table of n, a(n) for n = 0..1000</a>
%F A027277 From _Peter Luschny_, May 14 2016: (Start)
%F A027277 a(n) = hypergeom([1/2, -n, 1/2-n], [1, -2*n], -16) for n>=1.
%F A027277 a(n) = (2*n*(4*n-5)*(-9+4*n)*(-7+4*n)*a(n-3) - (4*n-5)*(50*n^3-175*n^2+152*n-9)* a(n-2) + (80*n^3-260*n^2+198*n-27)*(n-1)*a(n-1)) / (n*(n-1)*(-9+4*n)*(-1+2*n)) for n>=3. (End)
%F A027277 a(n) ~ sqrt(5 + 13/sqrt(17)) * ((9 + sqrt(17))/2)^n / (4*sqrt(Pi*n)). - _Vaclav Kotesovec_, May 14 2016
%p A027277 a := n -> add(binomial(2*k,k)*binomial(2*n-k,k), k=0..n):
%p A027277 seq(a(n),n=0..23); # _Peter Luschny_, May 14 2016
%t A027277 Table[Sum[Binomial[2k, k] Binomial[2n-k, k], {k,0,n}], {n,0,30}] (* _Michael De Vlieger_, May 14 2016 *)
%o A027277 (PARI) vector(30, n, n--; b=binomial; sum(k=0,n, b(2*k,k)*b(2*n-k,k)) ) \\ _G. C. Greubel_, May 23 2017, modified Aug 03 2019
%o A027277 (Magma) B:=Binomial; [(&+[B(2*k,k)*B(2*n-k,k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Aug 03 2019
%o A027277 (Sage) b=binomial; [sum(b(2*k,k)*b(2*n-k,k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Aug 03 2019
%o A027277 (GAP) B:=Binomial;; List([0..30], n-> Sum([0..n], k-> B(2*k,k)*B(2*n-k,k) )); # _G. C. Greubel_, Aug 03 2019
%Y A027277 Cf. A026568.
%K A027277 nonn
%O A027277 0,2
%A A027277 _Clark Kimberling_
%E A027277 New name from _Peter Luschny_, May 14 2016