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A188387 Central coefficient in (1 + (2^n+1)*x + 2^n*x^2)^n for n>=0.

%I A188387 #17 Mar 16 2023 12:23:31
%S A188387 1,3,33,1161,140545,63148833,111254837505,793938286762113,
%T A188387 23282575640347295745,2812444483776375381074433,
%U A188387 1393909730376211388561041231873
%N A188387 Central coefficient in (1 + (2^n+1)*x + 2^n*x^2)^n for n>=0.
%H A188387 Vincenzo Librandi, <a href="/A188387/b188387.txt">Table of n, a(n) for n = 0..36</a>
%F A188387 a(n) = Sum_{k=0..n} C(n,k)^2 * 2^(n*k).
%F A188387 a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Feb 12 2015
%t A188387 Table[Sum[Binomial[n,k]^2 * 2^(n*k), {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Feb 11 2015 *)
%o A188387 (PARI) {a(n)=polcoeff((1+(2^n+1)*x+2^n*x^2+x*O(x^n))^n,n)}
%o A188387 (PARI) {a(n)=sum(k=0,n,binomial(n,k)^2*2^(n*k))}
%o A188387 (Magma) /*1*/ P<x>:=PolynomialRing(Integers()); [ Coefficients((1+(2^n+1)*x+2^n*x^2)^n)[n+1]: n in [0..10] ]; /*2*/ &cat[ [&+[ Binomial(n, k)^2*2^(n*k): k in [0..n]]]: n in [0..10] ]; // _Bruno Berselli_, Mar 30 2011
%Y A188387 Cf. A187021.
%K A188387 nonn
%O A188387 0,2
%A A188387 _Paul D. Hanna_, Mar 29 2011