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A006134 a(n) = Sum_{k=0..n} binomial(2*k,k).
(Formerly M2811)
65
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

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

REFERENCES

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

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, #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, 2016, 4, 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(1996), no. 2, 83--89. MR1418955 (97k:33004)

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

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;

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

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 *)

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

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

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

CROSSREFS

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

Cf. A006135, A006136, A045912.

Equals A066796 + 1.

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

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Last modified December 5 08:40 EST 2022. Contains 358585 sequences. (Running on oeis4.)