A006134 -id:A006134 - cp's OEIS frontend

A167859 a(n) = 4^n * Sum_{k=0..n} binomial(2*k, k)^2 / 4^k.

1, 8, 68, 672, 7588, 93856, 1229200, 16695424, 232418596, 3293578784, 47309094672, 686870685312, 10059942413584, 148412250014336, 2202990595617344, 32873407393419776, 492791264816231204

Offset: 0

Alexander Adamchuk, Nov 13 2009

Every a(n) from a((p-1)/2) to a(p-1) is divisible by prime p for p = {7, 47, 191, 383, 439, 1151, 1399, 2351, 2879, 3119, 3511, 3559, ...} = A167860, apparently a subset of primes of the form 8n+7 (A007522).
7^3 divides a(13) and 7^2 divides a(10)-a(13).
Every a(n) from a(kp-1 - (p-1)/2) to a(kp-1) is divisible by prime p from A167860.
Every a(n) from a((p^2-1)/2) to a(p^2-1) is divisible by prime p from A167860. For p=7 every a(n) from a((p^3-1)/2) to a(p^3-1) and from a((p^4-1)/2) to a(p^4-1)is divisible by p^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167860, A007522.

A167859 := proc(n)
    add( (binomial(2*k,k)/2^k)^2,k=0..n) ;
    4^n*% ;
end proc:
seq(A167859(n),n=0..20) ; # R. J. Mathar, Sep 21 2016

Table[4^n*Sum[Binomial[2*k,k]^2/4^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n) = 4^n*sum(k=0,n, binomial(2*k,k)^2/4^k) \\ Charles R Greathouse IV, Sep 21 2016

Recurrence: n^2*a(n) = 4*(5*n^2 - 4*n + 1)*a(n-1) - 16*(2*n - 1)^2*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 2^(4*n+2)/(3*Pi*n). - Vaclav Kotesovec, Oct 20 2012
G.f.: 2*EllipticK(4*sqrt(x))/(Pi*(1-4*x)), where EllipticK is the complete elliptic integral of the first kind, using the Gradshteyn and Ryzhik convention, also used by Maple.  In the convention of Abramowitz and Stegun, used by Mathematica, this would be written as 2*K(16*x)/(Pi*(1-4*x)). - Robert Israel, Sep 21 2016

More terms from Sean A. Irvine, Apr 14 2010

Further terms from Jon E. Schoenfield, May 09 2010

A360153 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-6k,n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 72, 258, 945, 3504, 13128, 49565, 188260, 718560, 2753721, 10588860, 40835160, 157871241, 611669250, 2374441380, 9233006541, 35956933050, 140220970200, 547490880981, 2140055896770, 8373651697800, 32795094564081, 128550662334522

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A105872, A144904, A360150, A360151, A360152, A360168.

Cf. A006134, A106188.

A360153 := proc(n)
    add(binomial(2*n-6*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360153(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a[n_] := Sum[Binomial[2*n - 6*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)

a(n) = sum(k=0, n\3, binomial(2*n-6*k, n-3*k));

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3) ).
a(n) ~ 2^(2*n + 6) / (63 * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
a(n)-a(n-3) = A000984(n). - R. J. Mathar, Mar 12 2023
D-finite with recurrence n*a(n) +2*(-2*n+1)*a(n-1) -n*a(n-3) +2*(2*n-1)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A059984 Concatenation of Łukasiewicz words.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 0, 0, 2, 1, 0, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 3, 0, 1, 1, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 1, 0, 0, 0, 3, 1, 0, 1, 1, 0, 2, 0, 0, 2, 0, 2, 0, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 0, 0, 2, 2, 0, 1, 0, 0, 3, 0, 0, 1, 0, 3, 0, 0, 0, 1, 3, 0, 0, 0, 0, 4, 0, 1, 1, 1, 1, 1

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Mar 07 2001

nonn

There are A000108(n-1) (Catalan numbers) Łukasiewicz words of length n.

The first occurrence of n in this sequence is a(A006134(n)-1).

			Łukasiewicz words: 0 01 011 002 0111 0021 0102 0012 0003 01111 00211 01021 00121 00031 01102 00202 01012 00112 00022 01003 00103 00013 00004 ...

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
Sean A. Irvine, Java program (github)
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz

Cf. A059985.

A078478 Number of rational knots of n crossings with signature 0 (chiral pairs counted twice).

Original entry on oeis.org

0, 1, 0, 3, 2, 9, 6, 29, 30, 99, 112, 351, 450, 1275, 1734, 4707, 6762, 17577, 26208, 66197, 101862, 250953, 395804, 956385, 1540110, 3660541, 5997600, 14061141, 23382294, 54177741, 91246662, 209295261, 356432166, 810375651, 1393592512

Offset: 3

Ralf Stephan, Jan 03 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
A. Stoimenow, Generating Functions, Fibonacci Numbers and Rational Knots

Cf. A006134.

CoefficientList[Series[(- x/2) (2 + (2*x^4 - x^2 - 1) / (Sqrt[1 - 4 x^4] (1 + x^2)) + (2 x^2 - x - 1) / (Sqrt[1 - 4 x^2] (1 + x))) / x^3, {x, 0, 40}], x] (* Vincenzo Librandi, May 17 2013 *)

G.f.: (-x/2)*( 2 + (2*x^4-x^2-1)/(sqrt(1-4*x^4)*(1+x^2)) + (2*x^2-x-1)/(sqrt(1-4*x^2)*(1+x)) ) - Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 01 2006

a(2n+4) = A006134(n-1) = Sum[ (2k)!/(k!)^2, {k,0,n} ]. - Alexander Adamchuk, Feb 23 2007

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 01 2006

A115255 "Correlation triangle" of central binomial coefficients A000984.

Original entry on oeis.org

1, 2, 2, 6, 5, 6, 20, 14, 14, 20, 70, 46, 41, 46, 70, 252, 160, 134, 134, 160, 252, 924, 574, 466, 441, 466, 574, 924, 3432, 2100, 1672, 1534, 1534, 1672, 2100, 3432, 12870, 7788, 6118, 5506, 5341, 5506, 6118, 7788, 12870, 48620, 29172, 22692, 20152, 19174

Offset: 0

Paul Barry, Jan 18 2006

Row sums are A033114. Diagonal sums are A115256. T(2n,n) is A115257. Corresponds to the triangle of antidiagonals of the correlation matrix of the sequence array for C(2n,n).

Let s=(1,2,6,20,...), (central binomial coefficients), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A115255 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203005 for characteristic polynomials of principal submatrices of M, with interlacing zeros. - Clark Kimberling, Dec 27 2011

			Triangle begins:
  1;
  2, 2;
  6, 5, 6;
  20, 14, 14, 20;
  70, 46, 41, 46, 70;
  252, 160, 134, 134, 160, 252;
Northwest corner (square format):
  1    2    6    20    70
  2    5    14   46    160
  6    14   41   134   466
  20   46   134  441   1534

Cf. A203004, A203001, A202453.

s[k_] := Binomial[2 k - 2, k - 1];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]]; (* A115255 in square format *)
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A006134 *)
Table[m[1, j], {j, 1, 12}]     (* A000984 *)
Table[m[j, j], {j, 1, 12}]     (* A115257 *)
Table[m[j, j + 1], {j, 1, 12}] (* 2*A082578 *)
(* Clark Kimberling, Dec 27 2011 *)

G.f.: 1/(sqrt(1-4*x)*sqrt(1-4*x*y)*(1-x^2*y)) (format due to Christian G. Bower).

T(n, k) = Sum_{j=0..n} [j<=k]*C(2*k-2*j, k-j)*[j<=n-k]*C(2*n-2*k-2*j, n-k-j).

A167867 a(n) = 2^n * Sum_{k=0..n} binomial(2*k,k)^3 / 2^k.

Original entry on oeis.org

1, 10, 236, 8472, 359944, 16722896, 822334816, 42068907200, 2215884717400, 119364801362800, 6545334930678816, 364137834051739200, 20502307365808906816, 1166063313963833813632, 66893439680369963627264

Offset: 0

Alexander Adamchuk, Nov 14 2009

nonn

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), A167713 (B=16).

The expression a(n) = B^n*Sum_{ k=0..n } binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A079727, A167867, A167868, A167869, A167870, A167872.

Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859.

Table[2^n Sum[Binomial[2k,k]^3/2^k,{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Mar 26 2012 *)

a(n) = 2^n * Sum_{k=0..n} binomial(2*k,k)^3 / 2^k.
Recurrence: n^3*a(n) = 2*(33*n^3 - 48*n^2 + 24*n - 4)*a(n-1) - 16*(2*n-1)^3*a(n-2). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 2^(6*n+5)/(31*(Pi*n)^(3/2)). - Vaclav Kotesovec, Aug 13 2013

More terms from Sean A. Irvine, Apr 27 2010

A167868 a(n) = 3^n * Sum_{k=0..n} binomial(2*k,k)^3 / 3^k.

Original entry on oeis.org

1, 11, 249, 8747, 369241, 17110731, 840221217, 42944901219, 2260581606657, 121714776747971, 6671749658197129, 371062413164972955, 20887218937200347281, 1187720356043817041843, 68124474120573747125529

Offset: 0

Alexander Adamchuk, Nov 14 2009

nonn

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), A167713 (B=16).

The expression a(n) = B^n*Sum_{k=0..n} binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A079727, A167867, A167868, A167869, A167870, A167872.

Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859.

Table[3^n Sum[Binomial[2k,k]^3/3^k,{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 26 2012 *)

a(n) = 3^n * Sum_{k=0..n} binomial(2*k,k)^3 / 3^k.
Recurrence: n^3*a(n) = (67*n^3 - 96*n^2 + 48*n - 8)*a(n-1) - 24*(2*n-1)^3*a(n-2). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 2^(6*n+6)/(61*(Pi*n)^(3/2)). - Vaclav Kotesovec, Aug 13 2013

More terms from Sean A. Irvine, Apr 27 2010

A167869 a(n) = 4^n * Sum_{k=0..n} binomial(2*k,k)^3 / 4^k.

Original entry on oeis.org

1, 12, 264, 9056, 379224, 17519904, 858968640, 43860112128, 2307187351512, 124161781334048, 6803252453289408, 378260174003539200, 21287072393719585216, 1210206988807094340864, 69402141007670673363456

Offset: 0

Alexander Adamchuk, Nov 14 2009

nonn

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), A167713 (B=16).

The expression a(n) = B^n*Sum_{k=0..n} binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A079727, A167867, A167868, A167869, A167870, A167872.

Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859.

Table[4^n Sum[Binomial[2k,k]^3/4^k,{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Mar 26 2012 *)

a(n) = 4^n * Sum_{k=0..n} binomial(2*k,k)^3 / 4^k.
Recurrence: n^3*a(n) = 4*(17*n^3 - 24*n^2 + 12*n - 2)*a(n-1) - 32*(2*n-1)^3*a(n-2). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 2^(6*n+4)/(15*(Pi*n)^(3/2)). - Vaclav Kotesovec, Aug 13 2013

More terms from Sean A. Irvine, Apr 27 2010

A167870 a(n) = 16^n * Sum_{k=0..n} binomial(2*k,k)^3 / 16^k.

Original entry on oeis.org

1, 24, 600, 17600, 624600, 25996608, 1204834752, 59701593600, 3086972400600, 164324590337600, 8935798773354816, 494019944564058624, 27678350810730366400, 1567912312203901862400, 89647910047704725798400

Offset: 0

Alexander Adamchuk, Nov 14 2009

nonn

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), A167713 (B=16).

The expression a(n) = B^n*Sum_{k=0..n} binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A079727, A167867, A167868, A167869, A167870, A167872.

Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859.

Table[16^n Sum[Binomial[2k,k]^3/16^k,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 21 2012 *)

a(n) = 16^n * Sum_{k=0..n} binomial(2*k,k)^3 / 16^k.
Recurrence: n^3*a(n) = 8*(10*n^3 - 12*n^2 + 6*n - 1)*a(n-1) - 128*(2*n-1)^3*a(n-2). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 2^(6*n+2)/(3*(Pi*n)^(3/2)). - Vaclav Kotesovec, Aug 13 2013

More terms from Sean A. Irvine, Apr 27 2010

A167871 a(n) = 64^n * Sum_{k=0..n} binomial(2*k,k)^3 / 64^k.

Original entry on oeis.org

1, 72, 4824, 316736, 20614104, 1335305664, 86248451520, 5560325134848, 357992555533272, 23026456586057408, 1479999826835627328, 95071036081670530560, 6104320340924619384256, 391801560518407856592384

Offset: 0

Alexander Adamchuk, Nov 14 2009

nonn

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), A167713 (B=16).
The expression a(n) = B^n*Sum_{k=0..n} binomial(2*k,k)^3/B^k gives A079727 for B=1, A167867 (B=2), A167868 (B=3), A167869 (B=4), A167870 (B=16), A167871 (B=64).
p^2 divides all a(n) from n = (p-1)/2 to n = p-1 for prime p of the form p = 4k+3, p = {3,7,11,19,23,31,43,47,59,...} = A002145.
p^2 divides all a(n) from n = (2p-1 - (p-1)/2) to n = 2p-1 for prime p of the form p = 4k+3.
p^2 divides all a(n) from n = (3p-1 - (p-1)/2) to n = 3p-1 for prime p of the form p = 4k+3.
p^2 divides all a(n) from n = (p^2-1)/2 to n = p^2-1 for prime p of the form p = 4k+3.

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Wiley, 1968, p. 361.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A079727, A167867, A167868, A167869, A167870, A167872.

Cf. A000984, A066796, A006134, A082590, A132310, A002457, A144635, A167713, A167859, A002145.

Table[64^n Sum[Binomial[2k,k]^3/64^k,{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 26 2012 *)

a(n) = 64^n * Sum_{k=0..n} binomial(2*k,k)^3 / 64^k.
Recurrence: n^3*a(n) = 8*(4*n-1)*(4*n^2 - 2*n + 1)*a(n-1) - 512*(2*n-1)^3 *a(n-2). - Vaclav Kotesovec, Aug 14 2013
a(n) ~ 64^n*(Pi/GAMMA(3/4)^4 - 2/(Pi^(3/2)*sqrt(n))). - Vaclav Kotesovec, Aug 14 2013

More terms from Sean A. Irvine, Apr 25 2010

cp's OEIS Frontend

A167859 a(n) = 4^n * Sum_{k=0..n} binomial(2*k, k)^2 / 4^k.

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A360153 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-6*k,n-3*k).

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A059984 Concatenation of Łukasiewicz words.

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A078478 Number of rational knots of n crossings with signature 0 (chiral pairs counted twice).

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A115255 "Correlation triangle" of central binomial coefficients A000984.

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A167867 a(n) = 2^n * Sum_{k=0..n} binomial(2*k,k)^3 / 2^k.

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Mathematica

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A167868 a(n) = 3^n * Sum_{k=0..n} binomial(2*k,k)^3 / 3^k.

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Mathematica

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A167869 a(n) = 4^n * Sum_{k=0..n} binomial(2*k,k)^3 / 4^k.

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A360153 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-6k,n-3*k).