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A216584 a(n) = A002426(n)*A000984(n); product of central trinomial coefficients and central binomial coefficients.

%I A216584 #24 May 13 2025 12:07:39
%S A216584 1,2,18,140,1330,12852,130284,1348776,14247090,152618180,1654120468,
%T A216584 18096447096,199536967084,2214714164600,24720932068200,
%U A216584 277289164574640,3123590583844530,35318969120870820,400692715550057700,4559427798654821400,52020436064931914580
%N A216584 a(n) = A002426(n)*A000984(n); product of central trinomial coefficients and central binomial coefficients.
%H A216584 G. C. Greubel, <a href="/A216584/b216584.txt">Table of n, a(n) for n = 0..925</a>
%F A216584 a(n) = binomial(2*n, n) * Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k, k).
%F A216584 Logarithmic derivative of A216585, after ignoring initial term a(0).
%F A216584 a(n) = [x^n*y^n] ( 1 + (x + y)^2 + (x + y)^4 )^n. - _Peter Bala_, Feb 17 2020
%F A216584 G.f.: hypergeom([1/2, 1/2],[1],16*x/(1+4*x))/sqrt(1+4*x). - _Mark van Hoeij_, May 13 2025
%e A216584 L.g.f.: L(x) = 2*x + 18*x^2/2 + 140*x^3/3 + 1330*x^4/4 + 12852*x^5/5 + 130284*x^6/6 + ...
%e A216584 where
%e A216584 exp(L(x)) = 1 + 2*x + 11*x^2 + 66*x^3 + 485*x^4 + 3842*x^5 + 32712*x^6 + ... + A216585(n)*x^n/n + ...
%e A216584 The central trinomial coefficients (A002426) begin:
%e A216584 [1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, ...];
%e A216584 The central binomial coefficients (A000984) begin:
%e A216584 [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, ...].
%t A216584 Table[Binomial[2*n, n]*Sum[ Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}], {n,0,50}] (* _G. C. Greubel_, Feb 27 2017 *)
%o A216584 (PARI) {a(n) = polcoeff((1+x+x^2)^n,n) * polcoeff((1+2*x+x^2)^n,n)}
%o A216584 (PARI) {a(n)=binomial(2*n,n)*sum(k=0,n\2,binomial(n,2*k)*binomial(2*k,k))}
%o A216584 for(n=0,21,print1(a(n),", "))
%Y A216584 Cf. A216585, A151341, A168595, A002426, A000984.
%K A216584 nonn
%O A216584 0,2
%A A216584 _Paul D. Hanna_, Sep 08 2012