This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A033434 #37 Jul 10 2023 03:04:53
%S A033434 1,4,13,43,145,497,1727,6071,21554,77180,278426,1010990,3692213,
%T A033434 13553555,49981875,185082495,687923790,2565602160,9598056630,
%U A033434 36008860650,135446603370,510706730274,1929930236790,7308166696118,27727426756580,105387411817352,401231661076148,1529970156473276,5842655231153741,22342874048993015
%N A033434 Third differences of Catalan numbers A000108.
%H A033434 Vincenzo Librandi, <a href="/A033434/b033434.txt">Table of n, a(n) for n = 0..200</a>
%H A033434 Jocelyn Quaintance and Harris Kwong, <a href="https://www.emis.de/journals/INTEGERS/papers/n29/n29.Abstract.html">A combinatorial interpretation of the Catalan and Bell number difference tables</a>, Integers, 13 (2013), #A29.
%F A033434 a(n) = ( 27*n^3 + 81*n^2 + 108*n + 24 )*binomial(2*n, n)/( (n+1)*(n+2)*(n+3)*(n+4) ). - _Benoit Cloitre_, Jun 11 2004
%F A033434 a(n) = -binomial(2*n,n)/(n+1)*hypergeom([-3,n+1/2],[n+2],4). - _Peter Luschny_, Aug 15 2012
%F A033434 G.f.: (1 + x + x^2*C(x)^3)*C(x)^3 where C(x) is the g.f. of A000108. - _Philippe Deléham_, Feb 04 2014
%F A033434 From _G. C. Greubel_, May 03 2021: (Start)
%F A033434 G.f.: (x + (1-x)*C(x))*C(x)^3, where C(x) is the g.f. of A000108.
%F A033434 E.g.f.: exp(2*x)*(BesselI(0, 2*x) +2*BesselI(1, 2*x) -BesselI(2, 2*x) -BesselI(3, 2*x) - BesselI(4, 2*x)).
%F A033434 a(n) = Sum_{k=0..3} (-1)^k*binomial(3,k)*C(n-k+3), where C(n) = A000108(n). (End)
%F A033434 Sum_{n>=0} a(n)/4^n = 14. - _Amiram Eldar_, Jul 10 2023
%p A033434 C:= n-> binomial(2*n,n)/(n+1);
%p A033434 a:= n-> add((-1)^j*binomial(3,j)*C(n-j+3), j=0..3);
%p A033434 seq(a(n), n=0..30); # _G. C. Greubel_, May 03 2021
%t A033434 Table[(27n^3 +81n^2 +108n +24)*n!*Binomial[2n, n]/(n+4)!, {n, 0, 40}] (* _Vincenzo Librandi_, Feb 05 2014 *)
%t A033434 Differences[CatalanNumber[Range[0,40]],3] (* _Harvey P. Dale_, Jul 05 2020 *)
%o A033434 (PARI) a(n)=( 27*n^3 + 81*n^2 + 108*n + 24)*n!*binomial(2*n,n)/(n+4)!;
%o A033434 (Magma) [(27*n^3+81*n^2+108*n+24)*Binomial(2*n, n)/((n+1)*(n+2)*(n+3)*(n+4)): n in [0..30]]; // _Vincenzo Librandi_, Feb 05 2014
%o A033434 (Sage) [sum((-1)^j*binomial(3,j)*catalan_number(n-j+3) for j in (0..3)) for n in (0..40)] # _G. C. Greubel_, May 03 2021
%Y A033434 Cf. A000108.
%K A033434 nonn
%O A033434 0,2
%A A033434 _N. J. A. Sloane_