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
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)
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
Max A. Alekseyev, Tewodros Amdeberhan, Jeffrey Shallit, and Ingrid Vukusic, On the p-adic valuations of values of Legendre polynomials, Res. Num. Theory 12 (2026), Art. 52. See also arXiv:2505.08935 [math.NT], 2025. See p. 3.
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.
Cody Baker, Moshe Cohen, Henry Dam, Rebecca Felber, Neal Madras, Ritvik Saha, and Daisy Thackrah, A central limit theorem for the signatures of 2-bridge knots, arXiv:2604.21107 [math.GT], 2026. See p. 5.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, J. Int. Seq. 19 (2016), Article 16.3.5.
Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq. 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. Fred Lunnon, The Pascal matrix, Fib. Quart. 15 (1977), 201-204.
Kim McInturff and Rob Pratt, Representations of a generating function, College Math. J. 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(2) (1996), 83-89. MR1418955 (97k:33004).
Mehtaab Sawhney, proposer, Problem 1102 Problems and Solutions, College Math. J. 48(3) (May 2017), 219-225, See p. 219; Radouan Boukharfane, A binomial identity, Solution to Problem 1102, ibid., 49(3) (May 2018), 225-226.
Wikipedia, Pascal Matrix.
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
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;
# Alternative:
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
# Alternative:
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 *)
(* Alternative: *)
a[ n_] := (4/3) Binomial[ 2 n, n] Hypergeometric2F1[ 1/2, 1, -n + 1/2, -1/3] (* Michael Somos, Jun 20 2012 *)
(* Alternative: *)
Accumulate[Table[Binomial[2n, n], {n, 0, 30}]] (* Harvey P. Dale, Jan 11 2015 *)
(* Alternative: *)
CoefficientList[Series[1/((1 - x) Sqrt[1 - 4 x]), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 13 2015 *)
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
(PARI) my(x='x+O('x^100)); Vec(1/((1-x)*sqrt(1-4*x))) \\ Altug Alkan, Oct 29 2015
(Maxima) makelist(sum(binomial(2*k, k), k, 0, n), n, 0, 12); /* Emanuele Munarini, Mar 15 2011 */
(Magma) &cat[ [&+[ Binomial(2*k, k): k in [0..n]]]: n in [0..30]]; // Vincenzo Librandi, Aug 13 2015
KEYWORD
nonn,changed
AUTHOR
EXTENSIONS
Simpler definition from Alexander Adamchuk, Jul 05 2006
STATUS
approved
