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A033434 Third differences of Catalan numbers A000108. 5
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 (list; graph; refs; listen; history; text; internal format)
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

MATHEMATICA

Table[(27 n^3 + 81 n^2 + 108 n + 24) Binomial[2 n, n]/(n + 1)/(n + 2)/(n + 3)/(n + 4), {n, 0, 40}] (* Vincenzo Librandi, Feb 05 2014 *)

PROG

(PARI) a(n)=( 27*n^3 + 81*n^2 + 108*n + 24)*binomial(2*n, n)/(n+1)/(n+2)/(n+3)/(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

CROSSREFS

Cf. A000108.

Sequence in context: A121486 A188176 A003688 * A297928 A113986 A149426

Adjacent sequences:  A033431 A033432 A033433 * A033435 A033436 A033437

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified September 15 08:13 EDT 2019. Contains 327062 sequences. (Running on oeis4.)