A006134 a(n) = Sum_{k=0..n} binomial(2*k,k).

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, 169719408596403, 665637941544507

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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

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

From Gus Wiseman, Apr 20 2023: (Start)

Also the number of nonempty subsets of {1,...,2n+1} with median n+1, where the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). The odd/even-length cases are A000984 and A006134(n-1). For example, the a(0) = 1 through a(2) = 9 subsets are:

{1} {2} {3}

{1,3} {1,5}

{1,2,3} {2,4}

{1,3,4}

{1,3,5}

{2,3,4}

{2,3,5}

{1,2,4,5}

{1,2,3,4,5}

Alternatively, a(n-1) is the number of nonempty subsets of {1,...,2n-1} with median n.

(End)

Examples

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

References

• Marko Petkovsek, Herbert Wilf and Doron Zeilberger, A=B, A K 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
• Moa Apagodu and Doron Zeilberger, Using the "Freshman's Dream" to Prove Combinatorial Congruences, arXiv:1606.03351 [math.CO], 2016. Also Amer. Math. Monthly. 124 (2017), 597-608.
• Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19 (2016), Article 16.3.5.
• Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• J. Hietarinta, T. Mase and R. Willox, Algebraic entropy computations for lattice equations: why initial value problems do matter, arXiv:1909.03232 [nlin.SI], 2019.
• Neelam J. Kumar, N-Summet-k and Its Application in the Construction of Pascal Triangle and Pascal Matrix, Journal of Applied Mathematics and Physics, 4 (2016), 169-177.
• W. F. Lunnon, The Pascal matrix, Fib. Quart., Vol. 15 (1977), pp. 201-204.
• Kim McInturff and Rob Pratt, Representations of a generating function, The College Mathematics Journal, 40 (2009), 294-296.
• Hao Pan and Zhi-Wei Sun, A combinatorial identity with application to Catalan numbers, arXiv:math/0509648 [math.CO], 2005-2006.
• Peter Paule, A proof of a conjecture of Knuth. Experiment. Math. 5, No. 2 (1996), 83-89. MR1418955 (97k:33004).
• Mehtaab Sawhney, proposer, Problem 1102 Problems and Solutions, The College Mathematics Journal, Vol. 48, No. 3 (May 2017), pp. 219-225, p. 219; Radouan Boukharfane, A binomial identity, Solution to Problem 1102, ibid., Vol. 49, No. 3 (May 2018), pp. 225-226.
• Wikipedia, Pascal Matrix.

Crossrefs

Cf. A000984 (first differences), A097933, A038874, A132310.

Cf. A006135, A006136, A045912.

Equals A066796 + 1.

Odd bisection of A100066.

Row sums of A361654 (also column k = 2).

A007318 counts subsets by length, A231147 by median, A013580 by integer median.

A359893 and A359901 count partitions by median.

Cf. A000975, A024718, A057552, A079309, A327481.

Programs

MATLAB

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

Magma

&cat[ [&+[ Binomial(2*k, k): k in [0..n]]]: n in [0..30]]; // Vincenzo Librandi, Aug 13 2015

Maple

A006134 := proc(n) sum(binomial(2*k,k),k=0..n); end;
a := n -> -binomial(2*(n+1),n+1)*hypergeom([1,n+3/2],[n+2], 4) - I/sqrt(3):
seq(simplify(a(n)), n=0..24); # Peter Luschny, Oct 29 2015
# third program:
A006134 := series(exp(2*x)*BesselI(0, 2*x) + exp(x)*int(BesselI(0, 2*x)*exp(x), x), x = 0, 25):
seq(n!*coeff(A006134, x, n), n=0..24); # Mélika Tebni, Feb 27 2024

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 *)
Accumulate[Table[Binomial[2n,n],{n,0,30}]] (* Harvey P. Dale, Jan 11 2015 *)
CoefficientList[Series[1/((1 - x) Sqrt[1 - 4 x]), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 13 2015 *)

Maxima

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

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

PARI

my(x='x+O('x^100)); Vec(1/((1-x)*sqrt(1-4*x))) \\ Altug Alkan, Oct 29 2015

Formula

From Alexander Adamchuk, Jul 05 2006: (Start)

a(n) = Sum_{k=0..n} (2k)!/(k!)^2.

a(n) = A066796(n) + 1, n>0. (End)

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

D-finite with 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

From Alzhekeyev Ascar M, Jan 19 2012: (Start)

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).

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). (End)

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

a(n) = Sum_{k = 0..n} binomial(n+1,k+1)*A002426(k). - Peter Bala, Oct 29 2015

a(n) = -binomial(2*(n+1),n+1)*hypergeom([1,n+3/2],[n+2], 4) - i/sqrt(3). - Peter Luschny, Oct 29 2015

a(n) = binomial(2*n, n)*hypergeom([1,-n], [1/2-n], 1/4). - Peter Luschny, Mar 16 2016

From Gus Wiseman, Apr 20 2023: (Start)

a(n+1) - a(n) = A000984(n).

a(n) = A013580(2n+1,n+1) (conjectured).

a(n) = 2*A024718(n) - 1.

a(n) = A100066(2n+1).

a(n) = A231147(2n+1,n+1) (conjectured). (End)

a(n) = Sum_{k=0..floor(n/3)} 3^(n-3*k) * binomial(n-k,2*k) * binomial(2*k,k) (Sawhney, 2017). - Amiram Eldar, Feb 24 2024

From Mélika Tebni, Feb 27 2024: (Start)

Limit_{n -> oo} a(n) / A281593(n) = 2.

E.g.f.: exp(2*x)*BesselI(0,2*x) + exp(x)*integral( BesselI(0,2*x)*exp(x) ) dx. (End)

a(n) = [(x*y)^n] 1/((1 - (x + y))*(1 - x*y)). - Stefano Spezia, Feb 16 2025

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(2*n+1-k, n-2*k). - Michael Weselcouch, Jun 17 2025

a(n) = binomial(1+2*n, n)*hypergeom([1, (1-n)/2, -n/2], [-1-2*n, 2+n], 4). - Stefano Spezia, Jun 18 2025

Extensions

Simpler definition from Alexander Adamchuk, Jul 05 2006