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A014137 Partial sums of Catalan numbers (A000108). 286
1, 2, 4, 9, 23, 65, 197, 626, 2056, 6918, 23714, 82500, 290512, 1033412, 3707852, 13402697, 48760367, 178405157, 656043857, 2423307047, 8987427467, 33453694487, 124936258127, 467995871777, 1757900019101, 6619846420553, 24987199492705, 94520750408709, 358268702159069 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is also the result of applying the transformation on generating functions A -> 1/((1-x)*(1-x*A)) to the g.f. for the Catalan numbers.

p divides a(p)-3 for prime p=3 and p=7,13,19,31,37,43.. = A002476 (Primes of form 6n + 1). p^2 divides a(p^2)-3 for prime p>3. - Alexander Adamchuk, Jul 11 2006

Prime p divides a(p) for p = {2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ...} = A045309 Primes congruent to {0, 2} mod 3; and A045309 Primes p such that x^3 = n (integer) has only one solution mod p. Nonprime numbers n such that n divides a(n) are listed in A128287 = {1, 8, 133, ...}. - Alexander Adamchuk, Feb 23 2007

For p prime >=5, a(p-1) = 1 or -2 (mod p) according as p = 1 or -1 (mod 3) (see Pan and Sun link). For example, with p=5, a(p-1) = 23 = -2 (mod p). - David Callan, Nov 29 2007

Hankel transform is A010892(n+1). [From Paul Barry, Apr 24 2009]

Equals INVERTi transform of A000245: (1, 3, 9, 28,...). [From Gary W. Adamson, May 15 2009]

The subsequence of prime partial sums of Catalan numbers begins: a(1) = 2, a(4) = 23, a(6) = 197, a(16) = 48760367, see A121852. [From Jonathan Vos Post, Feb 10 2010]

REFERENCES

N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

I. Pak, Partition identities and geometric bijections. Proc. Amer. Math. Soc. 132 (2004), 3457-3462.

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

W. Chammam, F. Marcellán and R. Sfaxi, Orthogonal polynomials, Catalan numbers, and a general Hankel determinant evaluation, Linear Algebra Appl. (2011), in press

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

Hao Pan and Zhi-Wei Sun, A combinatorial identity with application to Catalan numbers

FORMULA

G.f.: (1-(1-4*x)^(1/2))/(2*x*(1-x)).

Sum_{i=1..n} c(i) = Sum_{i=1..n} C(2*i-2, i-1)/i = 1/(n-1)! * [ n^(n-2) +C(n, 2)*n^C(n-3, 1) +{8*C(n-4, 0) +19*C(n-4, 1) +24*C(n-4, 2) +14*C(n-4, 3) +3*C(n-4, 4)}*n^(n-4) +{18*C(n-5, 0) +82*C(n-5, 1) +229*C(n-5, 2) +323*C(n-5, 3) +244*C(n-5, 4) +95*C(n-5, 5) +15*C(n-5, 6)}*n^(n-5) +... +C(n-3, 0)*(n-1)! ] (where c() = Catalan numbers A000108). - André F. Labossière, May 17 2004

a(n) = Sum[(2k)!/(k!)^2/(k+1),{k,0,n}]. - Alexander Adamchuk, Jul 11 2006

Conjecture: (n+1)*a(n)+(1-5n)*a(n-1) +2*(2n-1)*a(n-2)=0. - R. J. Mathar, Dec 14 2011

Let C(n+1) = binomial(2*n+2,n+1)/(n+2) and H(n) = hypergeometric([1,n+3/2],[n+3],4) then A014137(n) = -(-1)^(2/3)- C(n+1)*H(n) and A014138(n) = -I^(2/3)- C(n+1)*H(n). - Peter Luschny, Aug 09 2012

G.f. (conjecture): Q(0)/(1-x), where Q(k)= 1+(4*k+1)*x/(k+1-2*x*(k+1)*(4*k+3)/(2*x*(4*k+3)+(2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013

a(n) ~ 2^(2*n+2)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 10 2013

MAPLE

a:= proc(n) option remember; `if`(n<2, n+1,

      ((5*n-1)*a(n-1)-(4*n-2)*a(n-2))/(n+1))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, May 18 2013

MATHEMATICA

Table[Sum[(2k)!/(k!)^2/(k+1), {k, 0, n}], {n, 1, 30}] (* Alexander Adamchuk, Jul 11 2006 *)

Accumulate[CatalanNumber[Range[0, 30]]] (* Harvey P. Dale, May 08 2012 *)

PROG

(PARI) Vec((1-(1-4*x)^(1/2))/(2*x*(1-x))+O(x^99)) \\ Charles R Greathouse IV, Feb 11 2011

(PARI)

sm(v)={my(s=vector(#v)); s[1]=v[1]; for(n=2, #v, s[n]=v[n]+s[n-1]); s; }

C(n)=binomial(2*n, n)/(n+1);

sm(vector(66, n, C(n-1)))

/* Joerg Arndt, May 04 2013 */

CROSSREFS

a(n) = A014138(n-1)+1.

Cf. A000108, A094638, A001246, A033536, A000984, A094639, A006134, A082894, A002897, A079727, A002476, A045309, A128287.

Cf. A000245 [From Gary W. Adamson, May 15 2009]

Sequence in context: A000083 A092668 A164039 * A245158 A245159 A245160

Adjacent sequences:  A014134 A014135 A014136 * A014138 A014139 A014140

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 25 23:44 EST 2014. Contains 250017 sequences.