cp's OEIS Frontend

A134757 A007318 * A100071.

%I A134757 #22 May 28 2024 06:48:31
%S A134757 1,3,11,37,123,401,1293,4131,13107,41353,129873,406319,1267093,
%T A134757 3940431,12224579,37845117,116944371,360771417,1111332129,3418840431,
%U A134757 10504903809,32242682787,98863833159,302863592073,927025884477,2835306153351,8665554849903
%N A134757 A007318 * A100071.
%C A134757 Also A007318^(-1) * A037965. - _Gary W. Adamson_, Nov 10 2007
%H A134757 G. C. Greubel, <a href="/A134757/b134757.txt">Table of n, a(n) for n = 1..1000</a>
%F A134757 Binomial transform of A100071 starting [1, 2, 6, 12, 30, ...].
%F A134757 Inverse binomial transform of A037965 starting [1, 4, 18, 80, 350, ...].
%F A134757 a(n) = (n-1)! * [x^(n-1)] exp(x)*((1 + 2*x)*BesselI(0, 2*x) + 2*x*BesselI(1, 2*x)) for n>0, a(0) = 0. - _Peter Luschny_, Aug 26 2012
%F A134757 D-finite with recurrence (n-1)*a(n) = 3*(n-1)*a(n-1) +(n+1)*a(n-2) -3*(n-3)*a(n-3). - _R. J. Mathar_, Nov 09 2021
%F A134757 G.f.: x*(1-x)/((1-3*x)*sqrt((1+x)*(1-3*x))). - _G. C. Greubel_, May 28 2024
%e A134757 a(3) = 11 = (1, 2, 1) dot (1, 2, 6) = (1 + 4 + 6), where A100071 = (1, 2, 6, 12, 30, ...).
%e A134757 a(3) = 11 = (1, -2, 1) dot (1, 4, 18) = (1 - 8 + 18). - _Gary W. Adamson_, Nov 10 2007
%t A134757 a[n_]:= a[n]= Sum[(-1)^(n-k-1)*Binomial[n-1,k]*(k+1)*Binomial[2*k, k], {k,0,n-1}];
%t A134757 Table[a[n], {n,40}] (* _G. C. Greubel_, May 28 2024 *)
%o A134757 (Magma)
%o A134757 A134757:= func< n | (&+[(-1)^(n-k-1)*(k+1)^2*Binomial(n-1,k)*Catalan(k) : k in [0..n-1]]) >;
%o A134757 [A134757(n): n in [1..40]]; // _G. C. Greubel_, May 28 2024
%o A134757 (SageMath)
%o A134757 def A134757(n): return sum((-1)^(n-k-1)*(k+1)*binomial(n-1,k)*binomial( 2*k, k) for k in range(n))
%o A134757 [A134757(n) for n in range(1,41)] # _G. C. Greubel_, May 28 2024
%Y A134757 Cf. A037965, A100071, A107965.
%K A134757 nonn
%O A134757 1,2
%A A134757 _Gary W. Adamson_, Nov 08 2007