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%I #29 Sep 08 2022 08:45:32
%S 1,2,5,14,40,116,344,1047,3273,10500,34503,115838,396244,1377221,
%T 4851665,17285662,62173297,225424527,822919439,3021713140,11151957809,
%U 41340655956,153853915410,574593145517,2152679745351,8087904580883,30466311814036,115036597198845
%N Expansion of g.f. (1-z)^2*(1-sqrt(1-4*z))/(2*z*(1 - 3*z + 2*z^2 - z^3)).
%C Previous name was: Transform of A000108 by the T_{0,0} transformation (see link).
%H G. C. Greubel, <a href="/A136304/b136304.txt">Table of n, a(n) for n = 0..1000</a>
%H Richard Choulet, <a href="https://www.apmep.fr/IMG/pdf/curtz1.pdf">Curtz-like transformation</a>.
%F G.f.: (1-z)^2*(1-sqrt(1-4*z))/(2*z*(1 - 3*z + 2*z^2 - z^3)).
%F Conjecture: (n+1)*a(n) + (-8*n+1)*a(n-1) + 3*(7*n-8)*a(n-2) + (-23*n+49)*a(n-3) + (13*n-32)*a(n-4) + 2*(-2*n+7)*a(n-5) = 0. - _R. J. Mathar_, Feb 29 2016
%F a(n) = Sum_{j=0..n} A034943(j+1)*A000108(n-j). - _G. C. Greubel_, Apr 19 2021
%t A034943[n_]:= A034943[n]= Sum[Binomial[n+k-1, 3*k], {k, 0, n/2}];
%t Table[Sum[A034943[j+1]*CatalanNumber[n-j], {j,0,n}], {n,0,35}] (* _G. C. Greubel_, Apr 19 2021 *)
%o (Magma)
%o A034943:= func< n | (&+[Binomial(n+j-1, 3*j): j in [0..Floor(n/2)]]) >;
%o [(&+[A034943(j+1)*Catalan(n-j): j in [0..n]]): n in [0..35]]; // _G. C. Greubel_, Apr 19 2021
%o (Sage)
%o def A034943(n): return sum(binomial(n+j-1,3*j) for j in (0..n//2))
%o [sum(A034943(j+1)*catalan_number(n-j) for j in (0..n)) for n in (0..35)] # _G. C. Greubel_, Apr 19 2021
%Y Cf. A097550, A135364, A136302, A136303, A136305, A137229, A137234, A137249.
%Y Cf. A000108, A034943.
%K nonn,easy
%O 0,2
%A _Richard Choulet_, Mar 22 2008
%E New name using g.f., _Joerg Arndt_, Apr 20 2021
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