A066796 -id:A066796 - cp's OEIS frontend

A082406 Numbers k such that k divides Sum_{j=1..k} binomial(2j,j).

1, 2, 5, 8, 11, 12, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 130, 131, 137, 149, 167, 173, 179, 191, 196, 197, 227, 233, 238, 239, 251, 257, 263, 266, 269, 281, 293, 311, 317, 322, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479

Offset: 1

Benoit Cloitre, Apr 23 2003

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..4973
Vaclav Kotesovec, Plot of a(n)/(n*log(n)) for n = 2..10000

Cf. A066796.

Select[Range[500],Divisible[Sum[Binomial[2k,k],{k,#}],#]&] (* Harvey P. Dale, Feb 16 2013 *)
A066796 = Accumulate[Table[Binomial[2*k, k], {k, 1, 1000}]]; Select[Range[Length[A066796]], Divisible[A066796[[#]], #] &] (* Vaclav Kotesovec, Feb 15 2019 *)

Is a(n) asymptotic to c*n*log(n) with 2 < c < 2.3?

A120278 a(n) = Sum_{m=1..n} Sum_{k=1..m} C(2k,k), where C(2k,k) = (2*k)!/(k!)^2 = A000984(k).

Original entry on oeis.org

2, 10, 38, 136, 486, 1760, 6466, 24042, 90238, 341190, 1297574, 4958114, 19019254, 73196994, 282492254, 1092867904, 4236849774, 16455966944, 64020347914, 249431257704, 973100041934, 3800867789884, 14862066265434, 58170868424084

Offset: 1

Alexander Adamchuk, Jul 04 2006

a(2*(p-1)) is divisible by p^2 for p=7,13,19,31,37,43,61,67.. A002476 (Primes of the form 6m + 1).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000984, A066796, A002476.

Cf. A066796, A079309.

Table[Sum[Sum[(2k)!/(k!)^2,{k,1,m}],{m,1,n}],{n,1,50}]
CoefficientList[Series[(1/Sqrt[1-4 x]-1)/((x-1)^2 x),{x,0,50}],x] (* Harvey P. Dale, May 24 2011 *)

a(n) = Sum_{m=1..n} Sum_{k=1..m} (2*k)!/(k!)^2.
a(n) = 2 * Sum_{k=1..n} A079309(k) = Sum_{k=1..n} A066796(k). - Alexander Adamchuk, Sep 01 2006
G.f.: x*(1/sqrt(1-4*x)-1)/(x*(x-1)^2). - Harvey P. Dale, May 24 2011
Recurrence: n*a(n) = 2*(3*n-1)*a(n-1) - (9*n-4)*a(n-2) + 2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+4)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012

A147291 a(n) = Sum_{k=1..n^2-1} binomial(2k,k).

Original entry on oeis.org

0, 28, 17576, 209295260, 43308802158650, 150315393336149895056, 8610524734277600186228691452, 8068213695203463278728832778415607708, 122985780058082302876789680971972469134558550878, 30386103720799858392019761983012781659021124133753353112778

Offset: 1

N. J. A. Sloane, Apr 25 2009

nonn

D. Callan, Divisibility of a Central Binomial Sum: A11292 and A11307, Amer. Math. Monthly, 116 (2009), 468-470.

Cf. A146977, A066796, A147304.

Table[Sum[Binomial[2*k, k], {k, 1, n^2 - 1}], {n, 1, 10}] (* Vaclav Kotesovec, Jun 07 2019 *)

a(n) = sum(k=1, n^2-1, binomial(2*k,k)); \\ Michel Marcus, Jul 05 2018

a(n) ~ 4^(n^2) / (3*sqrt(Pi)*n). - Vaclav Kotesovec, Jun 07 2019

A167912 a(n) = (1/(3^n)^2) * Sum_{k=0..(3^n-1)} binomial(2k,k).

Original entry on oeis.org

1, 217, 913083596083, 18744974860247264575032720770000376335095039

Offset: 1

Alexander Adamchuk, Nov 15 2009

nonn

Note that a(n) mod 27 = a(n) mod 9 = a(n) mod 3 = 1.

The Maple program yields the first seven terms; easily adjustable for obtaining more terms. However, a(4) has 44 digits, a(5) has 140 digits, a(6) has 432 digits and a(7) has 1308 digits. - Emeric Deutsch, Nov 22 2009

Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Eric Weisstein's World of Mathematics, Binomial Sums.

Cf. A006134, A083096, A066796, A083097, A081601, A010060, A122485.

a := proc (n) options operator, arrow: (sum(binomial(2*k, k), k = 0 .. 3^n-1))/3^(2*n) end proc: seq(a(n), n = 1 .. 7); # Emeric Deutsch, Nov 22 2009

Table[(1/3^n)^2 * Sum[Binomial[2 k, k], {k, 0, 3^n - 1}], {n, 1, 5}] (* G. C. Greubel, Jul 01 2016 *)

a(4) from Emeric Deutsch, Nov 22 2009

A151912 Expansion of (1-8x-8x^3)/(1-2x+4x^2)^2.

Original entry on oeis.org

1, -4, -28, -56, 32, 416, 832, -256, -4864, -9728, 2048, 51200, 102400, -16384, -507904, -1015808, 131072, 4849664, 9699328, -1048576, -45088768, -90177536, 8388608, 411041792, 822083584, -67108864, -3690987520, -7381975040

Offset: 0

Paul Barry, Mar 15 2008

2*a(n) is the Hankel transform of A066796.

Index entries for linear recurrences with constant coefficients, signature (4,-12,16,-16).

a(n)=2^n*A138341(n);

A181990 a(n) = Sum_{0 <= k <= m < p} (binomial(m, k)^(p-1))/p, where p is the n-th prime.

Original entry on oeis.org

3, 399, 12708885, 124515078454872901983423, 39212583445587381894247266262023061, 43487633454143579523135045521112077473364484383507327790688372131, 157851796824901989964381293031623545741924564754192453966085327785455257503133278729

Offset: 2

Alexander Adamchuk, Apr 04 2012

nonn

a(n) is a sum of all elements in the first p rows of Pascal's triangle each raised to the (p-1) power and divided by p, where p is the n-th prime.

For p = 3 and 7 (and their powers like 3, 9, 27, ... and 7, 49, ...) the sums of all elements in n = p^k top rows of Pascal's triangle each raised to the (n-1) = (p^k-1) power are divisible by n^2 = p^(2k) for all k > 0.

Eric Weisstein's World of Mathematics, Pascal's Triangle
Eric Weisstein's World of Mathematics, Binomial Sums

Cf. A007318, A006134, A083096, A066796, A083097, A081601, A010060, A122485, A167912.

Table[(Sum[Binomial[m, k]^(Prime[n] - 1), {m, 0, Prime[n] - 1}, {k, 0, m}])/Prime[n], {n, 2, 10}]

a(n) = my(p=prime(n)); sum(m=0, p-1, sum(k=0, m, binomial(m,k)^(p-1))/p); \\ Michel Marcus, Dec 03 2018

cp's OEIS Frontend

A082406 Numbers k such that k divides Sum_{j=1..k} binomial(2j,j).

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A120278 a(n) = Sum_{m=1..n} Sum_{k=1..m} C(2*k,k), where C(2*k,k) = (2*k)!/(k!)^2 = A000984(k).

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A147291 a(n) = Sum_{k=1..n^2-1} binomial(2k,k).

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Formula

A167912 a(n) = (1/(3^n)^2) * Sum_{k=0..(3^n-1)} binomial(2k,k).

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Programs

Maple

Mathematica

Extensions

A151912 Expansion of (1-8x-8x^3)/(1-2x+4x^2)^2.

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Author

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Formula

A181990 a(n) = Sum_{0 <= k <= m < p} (binomial(m, k)^(p-1))/p, where p is the n-th prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A120278 a(n) = Sum_{m=1..n} Sum_{k=1..m} C(2k,k), where C(2k,k) = (2*k)!/(k!)^2 = A000984(k).