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%I M2811 #159 Feb 27 2024 07:06:33
%S 1,3,9,29,99,351,1275,4707,17577,66197,250953,956385,3660541,14061141,
%T 54177741,209295261,810375651,3143981871,12219117171,47564380971,
%U 185410909791,723668784231,2827767747951,11061198475551,43308802158651
%N a(n) = Sum_{k=0..n} binomial(2*k,k).
%C 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
%C T(n+1,1) from table A045912 of characteristic polynomial of negative Pascal matrix. - _Michael Somos_, Jul 24 2002
%C 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
%C 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
%C First column of triangle A187887. - _Michel Marcus_, Jun 23 2013
%C From _Gus Wiseman_, Apr 20 2023: (Start)
%C 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:
%C {1} {2} {3}
%C {1,3} {1,5}
%C {1,2,3} {2,4}
%C {1,3,4}
%C {1,3,5}
%C {2,3,4}
%C {2,3,5}
%C {1,2,4,5}
%C {1,2,3,4,5}
%C Alternatively, a(n-1) is the number of nonempty subsets of {1,...,2n-1} with median n.
%C (End)
%D Marko Petkovsek, Herbert Wilf and Doron Zeilberger, A=B, A K Peters, 1996, p. 22.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006134/b006134.txt">Table of n, a(n) for n = 0..1000</a>
%H Moa Apagodu and Doron Zeilberger, <a href="http://arxiv.org/abs/1606.03351">Using the "Freshman's Dream" to Prove Combinatorial Congruences</a>, arXiv:1606.03351 [math.CO], 2016. Also Amer. Math. Monthly. 124 (2017), 597-608.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19 (2016), Article 16.3.5.
%H Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%H J. Hietarinta, T. Mase and R. Willox, <a href="https://arxiv.org/abs/1909.03232">Algebraic entropy computations for lattice equations: why initial value problems do matter</a>, arXiv:1909.03232 [nlin.SI], 2019.
%H Neelam J. Kumar, <a href="http://dx.doi.org/10.4236/jamp.2016.41020">N-Summet-k and Its Application in the Construction of Pascal Triangle and Pascal Matrix</a>, Journal of Applied Mathematics and Physics, 4 (2016), 169-177.
%H W. F. Lunnon, <a href="http://www.fq.math.ca/Scanned/15-3/lunnon.pdf">The Pascal matrix</a>, Fib. Quart., Vol. 15 (1977), pp. 201-204.
%H Kim McInturff and Rob Pratt, <a href="http://www.jstor.org/stable/25653754">Representations of a generating function</a>, The College Mathematics Journal, 40 (2009), 294-296.
%H Hao Pan and Zhi-Wei Sun, <a href="http://arxiv.org/abs/math/0509648">A combinatorial identity with application to Catalan numbers</a>, arXiv:math/0509648 [math.CO], 2005-2006.
%H Peter Paule, <a href="http://projecteuclid.org/euclid.em/1047565639">A proof of a conjecture of Knuth</a>. Experiment. Math. 5, No. 2 (1996), 83-89. MR1418955 (97k:33004).
%H Mehtaab Sawhney, proposer, <a href="http://www.jstor.org/stable/10.4169/college.math.j.48.3.219">Problem 1102</a> Problems and Solutions, The College Mathematics Journal, Vol. 48, No. 3 (May 2017), pp. 219-225, p. 219; Radouan Boukharfane, <a href="https://www.jstor.org/stable/48661695">A binomial identity</a>, Solution to Problem 1102, ibid., Vol. 49, No. 3 (May 2018), pp. 225-226.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pascal_matrix">Pascal Matrix</a>.
%F From _Alexander Adamchuk_, Jul 05 2006: (Start)
%F a(n) = Sum_{k=0..n} (2k)!/(k!)^2.
%F a(n) = A066796(n) + 1, n>0. (End)
%F G.f.: 1/((1-x)*sqrt(1-4*x)).
%F 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
%F 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
%F From _Alzhekeyev Ascar M_, Jan 19 2012: (Start)
%F 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).
%F 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)
%F a(n) ~ 2^(2*n+2)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Nov 06 2012
%F 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
%F 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
%F a(n) = Sum_{k = 0..n} binomial(n+1,k+1)*A002426(k). - _Peter Bala_, Oct 29 2015
%F a(n) = -binomial(2*(n+1),n+1)*hypergeom([1,n+3/2],[n+2], 4) - i/sqrt(3). - _Peter Luschny_, Oct 29 2015
%F a(n) = binomial(2*n, n)*hypergeom([1,-n], [1/2-n], 1/4). - _Peter Luschny_, Mar 16 2016
%F From _Gus Wiseman_, Apr 20 2023: (Start)
%F a(n+1) - a(n) = A000984(n).
%F a(n) = A013580(2n+1,n+1) (conjectured).
%F a(n) = 2*A024718(n) - 1.
%F a(n) = A100066(2n+1).
%F a(n) = A231147(2n+1,n+1) (conjectured). (End)
%F 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
%F From _Mélika Tebni_, Feb 27 2024: (Start)
%F Limit_{n -> oo} a(n) / A281593(n) = 2.
%F E.g.f.: exp(2*x)*BesselI(0,2*x) + exp(x)*integral( BesselI(0,2*x)*exp(x) ) dx. (End)
%e 1 + 3*x + 9*x^2 + 29*x^3 + 99*x^4 + 351*x^5 + 1275*x^6 + 4707*x^7 + 17577*x^8 + ...
%p A006134 := proc(n) sum(binomial(2*k,k),k=0..n); end;
%p a := n -> -binomial(2*(n+1),n+1)*hypergeom([1,n+3/2],[n+2], 4) - I/sqrt(3):
%p seq(simplify(a(n)), n=0..24); # _Peter Luschny_, Oct 29 2015
%p # third program:
%p A006134 := series(exp(2*x)*BesselI(0, 2*x) + exp(x)*int(BesselI(0, 2*x)*exp(x), x), x = 0, 25):
%p seq(n!*coeff(A006134, x, n), n=0..24); # _Mélika Tebni_, Feb 27 2024
%t Table[Sum[((2k)!/(k!)^2),{k,0,n}], {n,0,50}] (* _Alexander Adamchuk_, Jul 05 2006 *)
%t a[ n_] := (4/3) Binomial[ 2 n, n] Hypergeometric2F1[ 1/2, 1, -n + 1/2, -1/3] (* _Michael Somos_, Jun 20 2012 *)
%t Accumulate[Table[Binomial[2n,n],{n,0,30}]] (* _Harvey P. Dale_, Jan 11 2015 *)
%t CoefficientList[Series[1/((1 - x) Sqrt[1 - 4 x]), {x, 0, 33}], x] (* _Vincenzo Librandi_, Aug 13 2015 *)
%o (MATLAB) n=10; x=pascal(n); trace(x)
%o (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
%o (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
%o (PARI) my(x='x+O('x^100)); Vec(1/((1-x)*sqrt(1-4*x))) \\ _Altug Alkan_, Oct 29 2015
%o (Maxima) makelist(sum(binomial(2*k,k),k,0,n),n,0,12); \\ _Emanuele Munarini_, Mar 15 2011
%o (Magma) &cat[ [&+[ Binomial(2*k, k): k in [0..n]]]: n in [0..30]]; // _Vincenzo Librandi_, Aug 13 2015
%Y Cf. A000984 (first differences), A097933, A038874, A132310.
%Y Cf. A006135, A006136, A045912.
%Y Equals A066796 + 1.
%Y Odd bisection of A100066.
%Y Row sums of A361654 (also column k = 2).
%Y A007318 counts subsets by length, A231147 by median, A013580 by integer median.
%Y A359893 and A359901 count partitions by median.
%Y Cf. A000975, A024718, A057552, A079309, A327481.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Simpler definition from _Alexander Adamchuk_, Jul 05 2006
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