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%I #22 Sep 08 2022 08:45:24
%S 1,2,7,22,72,234,763,2486,8099,26372,85833,279226,907946,2951066,
%T 9587981,31140034,101104048,328162170,1064856217,3454513274,
%U 11204337056,36332719182,117795920249,381848062066,1237615088203,4010710218384
%N Expansion of 1/(sqrt(1-2*x-3*x^2)*(2-M(x))), where M(x) is the g.f. of the Motzkin numbers A001006.
%C Binomial transform of A116383.
%C The substitution x-> x/(1+x+x^2) in the g.f. (this might be called an inverse Motzkin transform) yields the g.f. of A074331. - _R. J. Mathar_, Nov 10 2008
%H Vincenzo Librandi, <a href="/A116387/b116387.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n,j-k)*C(j,n-j).
%F Conjecture: n*(17*n-142)*a(n) + (17*n^2 + 95*n + 138)*a(n-1) + (-391*n^2 + 2488*n - 2908)*a(n-2) + (-17*n^2 - 603*n + 1892)*a(n-3) + 2*(697*n-2021)*(n-4)*a(n-4) + 60*(17*n-47)*(n-4)*a(n-5) = 0. - _R. J. Mathar_, Nov 15 2011
%F a(n) ~ (1+sqrt(5))^n * (5+sqrt(5)) / 10. - _Vaclav Kotesovec_, Feb 08 2014
%t Table[Sum[Binomial[n,j-k]Binomial[j,n-j],{k,0,n},{j,0,n}],{n,0,30}] (* _Harvey P. Dale_, Feb 08 2012 *)
%o (PARI) {a(n) = sum(k=0,n, sum(j=0,n, binomial(n, j-k)*binomial(j,n-j)))}; \\ _G. C. Greubel_, May 23 2019
%o (Magma) [(&+[ (&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]) : k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, May 23 2019
%o (Sage) [sum( sum(binomial(n, j-k)*binomial(j,n-j) for j in (0..n)) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, May 23 2019
%o (GAP) List([0..30], n-> Sum([0..n], k-> Sum([0..n], j-> Binomial(n, j-k)*Binomial(j, n-j) ))) # _G. C. Greubel_, May 23 2019
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 12 2006
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