A033434 Third differences of Catalan numbers A000108.

1, 4, 13, 43, 145, 497, 1727, 6071, 21554, 77180, 278426, 1010990, 3692213, 13553555, 49981875, 185082495, 687923790, 2565602160, 9598056630, 36008860650, 135446603370, 510706730274, 1929930236790, 7308166696118, 27727426756580, 105387411817352, 401231661076148, 1529970156473276, 5842655231153741, 22342874048993015

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Formula

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

a(n) = -binomial(2*n,n)/(n+1)*hypergeom([-3,n+1/2],[n+2],4). - Peter Luschny, Aug 15 2012

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

From G. C. Greubel, May 03 2021: (Start)

G.f.: (x + (1-x)*C(x))*C(x)^3, where C(x) is the g.f. of A000108.

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)).

a(n) = Sum_{k=0..3} (-1)^k*binomial(3,k)*C(n-k+3), where C(n) = A000108(n). (End)

Sum_{n>=0} a(n)/4^n = 14. - Amiram Eldar, Jul 10 2023

Maple

C:= n-> binomial(2*n, n)/(n+1);
a:= n-> add((-1)^j*binomial(3, j)*C(n-j+3), j=0..3);
seq(a(n), n=0..30); # G. C. Greubel, May 03 2021

Mathematica

Table[(27n^3 +81n^2 +108n +24)*n!*Binomial[2n, n]/(n+4)!, {n, 0, 40}] (* Vincenzo Librandi, Feb 05 2014 *)
Differences[CatalanNumber[Range[0, 40]], 3] (* Harvey P. Dale, Jul 05 2020 *)

Prog

(PARI) a(n)=( 27*n^3 + 81*n^2 + 108*n + 24)*n!*binomial(2*n, n)/(n+4)!;
(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
(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

Crossrefs

Cf. A000108.

Sequence in context: A339063 A188176 A003688 * A297928 A363366 A113986

Adjacent sequences: A033431 A033432 A033433 * A033435 A033436 A033437

Keyword

nonn

Author

N. J. A. Sloane

Status

approved