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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 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) 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 A113986 A149426 Adjacent sequences:  A033431 A033432 A033433 * A033435 A033436 A033437 KEYWORD nonn AUTHOR STATUS approved

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Last modified November 28 23:48 EST 2021. Contains 349415 sequences. (Running on oeis4.)