login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006134 a(n) = Sum_{ k=0..n } binomial(2*k,k).
(Formerly M2811)
44
1, 3, 9, 29, 99, 351, 1275, 4707, 17577, 66197, 250953, 956385, 3660541, 14061141, 54177741, 209295261, 810375651, 3143981871, 12219117171, 47564380971, 185410909791, 723668784231, 2827767747951, 11061198475551, 43308802158651 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)/B^k gives A006134 for B=1, A082590 (B=2), A132310 (B=3), A002457 (B=4), A144635 (B=5). - N. J. A. Sloane, Jan 21 2009

T(n+1,1) from table A045912 of characteristic polynomial of negative Pascal matrix. - Michael Somos, Jul 24 2002

p divides a((p-3)/2) for p=11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167..=A097933. Also primes congruent to {1, 2, 3, 11} mod 12 or primes p such that 3 is a square mod p (excluding 2 and 3) A038874. - Alexander Adamchuk, Jul 05 2006

Partial sums of the even central binomial coefficients. For p prime >=5, a(p-1) = 1 or -1 (mod p) according as p = 1 or -1 (mod 3) (see Pan and Sun link). - David Callan, Nov 29 2007

a(n)=Sum_{ k=0..n } b(k)*binomial(n+k,k), where b(k)=0 for n-k == 2 (mod 3), b(k)=1 for n-k == 0 or 1 (mod 6), and b(k)=-1 for n-k== 3 or 4 (mod 6). - Alzhekeyev Ascar M. - Jan 19 2012

a(n)=Sum_{ k=0..n-1 } c(k)*binomial(2n,k) + binomial(2n,n), where c(k)=0 for n-k == 0 (mod 3), c(k)=1 for n-k== 1 (mod 3), and c(k)=-1 for n-k==2 (mod 3). - Alzhekeyev Ascar M. - Jan 19 2012

First column of triangle A187887. - Michel Marcus, Jun 23 2013

REFERENCES

W. F. Lunnon, "The Pascal matrix", Fib. Quart. vol. 15 (1977) pp. 201-204.

Paule, Peter. A proof of a conjecture of Knuth. Experiment. Math. 5(1996), no. 2, 83--89. MR1418955 (97k:33004)

M. Petkovsek et al., A=B, Peters, 1996, p. 22.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Wikipedia, Pascal Matrix

FORMULA

a(n) = Sum[((2k)!/(k!)^2),{k,0,n}]. a(n) = A066796(n) + 1, n>0. - Alexander Adamchuk, Jul 05 2006

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

Recurrence: (n+2)*a(n+2) - (5*n+8)*a(n+1) + 2*(2*n+3)*a(n) = 0. [Emanuele Munarini, Mar 15 2011]

a(n) = C(2n,n) * Sum_{k=0..2n} (-1)^k*trinomial(n,k)/C(2n,k) where trinomial(n,k) = [x^k] (1 + x + x^2)^n. E.g. a(2) = C(4,2)*(1/1 - 2/4 + 3/6 - 2/4 + 1/1) = 6*(3/2) = 9 ; a(3) = C(6,3)*(1/1 - 3/6 + 6/15 - 7/20 + 6/15 - 3/6 + 1/1) = 20*(29/20) = 29. - Paul D. Hanna, Aug 21 2007

a(n) ~ 2^(2*n+2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 06 2012

G.f.: G(0)/2/(1-x), where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

G.f.: G(0)/(1-x), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2) - x*(4*k+2)*(4*k+3)/(x*(4*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013

EXAMPLE

1 + 3*x + 9*x^2 + 29*x^3 + 99*x^4 + 351*x^5 + 1275*x^6 + 4707*x^7 + 17577*x^8 + ...

MAPLE

A006134 := proc(n) sum(binomial(2*k, k), k=0..n); end;

MATHEMATICA

Table[Sum[((2k)!/(k!)^2), {k, 0, n}], {n, 0, 50}] (* Alexander Adamchuk, Jul 05 2006 *)

a[ n_] := (4/3) Binomial[ 2 n, n] Hypergeometric2F1[ 1/2, 1, -n + 1/2, -1/3] (* Michael Somos, Jun 20 2012 *)

PROG

(MATLAB) n=10; x=pascal(n); trace(x)

(PARI) {a(n) = if( n<0, 0, polcoeff( charpoly( matrix( n+1, n+1, i, j, -binomial( i+j-2, i-1))), 1))} /* Michael Somos, Jul 10 2002 */

(PARI) {a(n)=binomial(2*n, n)*sum(k=0, 2*n, (-1)^k*polcoeff((1+x+x^2)^n, k)/binomial(2*n, k))} - Paul D. Hanna, Aug 21 2007

(Maxima) makelist(sum(binomial(2*k, k), k, 0, n), n, 0, 12); [Emanuele Munarini, Mar 15 2011]

CROSSREFS

Cf. A000984, A066796, A097933, A038874.

Cf. A132310.

Cf. A006135, A006136, A045912. Differences give A000984.

Sequence in context: A151030 A066331 A099780 * A074526 A231291 A239116

Adjacent sequences:  A006131 A006132 A006133 * A006135 A006136 A006137

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Simpler definition from Alexander Adamchuk, Jul 05 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 19 21:57 EST 2014. Contains 252240 sequences.