|
%I #38 Jan 06 2023 15:46:57
%S 1,2,6,22,92,424,2108,11134,61748,356296,2123720,13002840,81417520,
%T 519550880,3369559864,22161337742,147544048324,992923683912,
%U 6746101933304,46226667046360,319199694771696,2219445498261152,15529758665102416,109291258152550712
%N Exponential convolution of Catalan numbers with themselves.
%H Vincenzo Librandi, <a href="/A014330/b014330.txt">Table of n, a(n) for n = 0..200</a>
%F From _Vladeta Jovovic_, Jan 01 2004: (Start)
%F E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x))^2.
%F a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k, k)/(k+1)*binomial(2*n-2*k, n-k)/(n-k+1).
%F a(n) = 4^n*Sum_{k=0..n} (-4)^(-k)*binomial(n, k)*binomial(k, floor(k/2))*binomial(k+1, floor((k+1)/2)).
%F a(n) = binomial(2*n, n)/(n+1)*hypergeometric3F2([-n-1, -n, 1/2], [2, 1/2-n], -1). (End)
%F (n + 1)*(n + 2)*a(n) = 4*(3*n^2 + n - 1)*a(n - 1) - 32*(n - 1)^2*a(n - 2). - _Vladeta Jovovic_, Jul 15 2004
%F a(n) = Sum_{k=0..n} binomial(n,k)*A000108(k)*A000108(n-k). - _Philippe Deléham_, Aug 23 2006
%F a(n) = (4*A053175(n) - A053175(n+1)/4) / ((n+2)*2^n). - _Mark van Hoeij_, Jul 02 2010
%F G.f.: (1-6*x)*hypergeometric2F1([1/2, 1/2],[2],16*x^2/(4*x-1)^2)/(2*x*(4*x-1)) - x*(8*x-1)*hypergeometric2F1([3/2, 3/2],[3],16*x^2/(4*x-1)^2)/(4*x-1)^3 + 1/(2*x). - _Mark van Hoeij_, Oct 25 2011
%F E.g.f.: hypergeometric1F1([1/2],[2],4*x)^2, coinciding with the above given e.g.f. - _Wolfdieter Lang_, Jan 13 2012
%F a(n) ~ 8^(n+1) / (Pi*n^3). - _Vaclav Kotesovec_, Feb 25 2014
%t Table[Sum[Binomial[n, k]*Binomial[2*k, k]/(k+1)*Binomial[2*n-2*k, n-k]/(n-k+1),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Feb 25 2014 *)
%o (PARI) A014330(n)=sum(k=0,n,binomial(n,k)*A000108(k)*A000108(n-k)) \\ _M. F. Hasler_, Jan 13 2012
%o (Magma)
%o A014330:= func< n | (&+[Binomial(n,k)*Catalan(k)*Catalan(n-k): k in [0..n]]) >;
%o [A014330(n): n in [0..40]]; // _G. C. Greubel_, Jan 06 2023
%o (SageMath)
%o def c(n): return catalan_number(n)
%o def A014330(n): return sum( binomial(n,k)*c(k)*c(n-k) for k in range(n+1))
%o [A014330(n) for n in range(41)] # _G. C. Greubel_, Jan 06 2023
%Y Cf. A000108, A014333, A053175, A126869, A138364.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vincenzo Librandi_, Feb 27 2014
|
