A066796 -id:A066796 - cp's OEIS frontend

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

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

A167860 Primes p dividing every A167859(m) from m=(p-1)/2 to m=(p-1).

Original entry on oeis.org

7, 47, 191, 383, 439, 1151, 1399, 2351, 2879, 3119, 3511, 3559, 4127, 5087, 5431, 6911, 8887, 9127, 9791, 9887, 12391, 13151, 14407, 15551, 16607, 19543, 20399, 21031, 21319, 21839, 23039, 25391, 26399, 28087, 28463, 28711, 29287, 33223, 39551, 43103, 44879, 46271

Offset: 1

Alexander Adamchuk, Nov 13 2009

nonn

Apparently A167860 is a subset of primes of the form 8*k + 7 (A007522).
Every A167859(m) from m=(p-1)/2 to m=(p-1) is divisible by prime p belonging to A167860.
7^3 divides A167859(13) and 7^2 divides A167859(10)-A167859(13).
Every A167859(m) from m=(kp-1 - (p-1)/2) to m=(kp-1) is divisible by prime p from A167860.
Every A167859(m) from m=((p^2-1)/2) to m=(p^2-1) is divisible by prime p from A167860. For p=7 every A167859(m) from m=((p^3-1)/2) to m=(p^3-1) and from m=((p^4-1)/2) to m(p^4-1)is divisible by p^2.

R. J. Mathar, Table of n, a(n) for n = 1..57

Cf. A000984, A006134, A007522, A066796, A082590, A132310, A144635, A167713, A167859, A276649.

A167859 := proc(n)
    option remember;
    if n <= 1 then
        add( (binomial(2*k, k)/2^k)^2, k=0..n) ;
        4^n*% ;
    else
        4*(5*n^2 - 4*n + 1)*procname(n-1) - 16*(2*n - 1)^2*procname(n-2) ;
        %/n^2 ;
    end if;
end proc:
isA167860 := proc(p)
    local m ;
    for m from (p-1)/2 to p-1 do
        if modp(A167859(m),p) > 0 then
            return false;
        end if;
    end do:
    true ;
end proc:
A167860 := proc(n)
    option remember ;
    if n = 0 then
        2;
    else
        p := nextprime(procname(n-1)) ;
        while not isA167860(p) do
            p := nextprime(p) ;
        end do ;
        return p;
    end if;
end proc:
seq(A167860(n),n=1..10) ; # R. J. Mathar, Jan 22 2025

is(p) = if(isprime(p)&&p%2, my(m=Mod(1, p), s=m); for(k=1, p\2, s+=(m*=(2*k-1)/k)^2); !s, 0); \\ Jinyuan Wang, Jul 24 2022

More terms from Jinyuan Wang, Jul 24 2022

A146977 a(n) = Sum_{k=1..prime(n)} binomial(2k,k).

Original entry on oeis.org

8, 28, 350, 4706, 956384, 14061140, 3143981870, 47564380970, 11061198475550, 40328534467494098, 624021469287616610, 2338958497948612884092, 568740383597968804344752, 8885203954833615367662872, 2175457720411301277807088352, 8390179243143767727343229863076

Offset: 1

N. J. A. Sloane, Apr 25 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..250

A subsequence of A066796.

[ &+[Binomial(2*k,k): k in [1..NthPrime(n)]]: n in [1..15]]; // G. C. Greubel, Jan 09 2020

seq( add(binomial(2*k,k), k=1..ithprime(n)), n=1..15); # G. C. Greubel, Jan 09 2020

Table[Sum[Binomial[2k,k],{k,Prime[n]}],{n,15}] (* Harvey P. Dale, Dec 31 2012 *)

vector(15, n, sum(k=1,prime(n), binomial(2*k,k)) ) \\ G. C. Greubel, Jan 09 2020

[sum(binomial(2*k,k) for k in (1..nth_prime(n))) for n in (1..15)] # G. C. Greubel, Jan 09 2020

A066798 a(n) = Sum_{i=1..n} binomial(6i,3i).

Original entry on oeis.org

20, 944, 49564, 2753720, 157871240, 9233006540, 547490880980, 32795094564080, 1979734520212192, 120244316085073616, 7339672750101339356, 449852213026938118560, 27666867082225970134160

Offset: 1

Benoit Cloitre, Jan 18 2002

nonn

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A006134, A066796, A066797, A066802.

s := RootOf((s+8)*s^3*x-s+1, s):
series( (1+8/s)^(3/2)*(s-4)*s^5/(3*(s^4+8*s^3-s+1)*(s^2+4*s-8)) - 1/(1-x), x=0, 30); # Mark van Hoeij, May 02 2013

Accumulate[Table[Binomial[6n,3n],{n,20}]] (* Harvey P. Dale, Apr 04 2020 *)

{ a=0; for (n=1, 100, write("b066798.txt", n, " ", a+=binomial(6*n, 3*n)) ) } \\ Harry J. Smith, Mar 28 2010

G.f.: (1+8/s)^(3/2)*(s-4)*s^5/(3*(s^4+8*s^3-s+1)*(s^2+4*s-8)) - 1/(1-x) where (s+8)*s^3*x-s+1 = 0. - Mark van Hoeij, May 02 2013
a(n) ~ sqrt(3) * 64^(n+1) / (189*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 07 2019
D-finite with recurrence n*(3*n-1)*(3*n-2)*a(n) +(-585*n^3+873*n^2-370*n+40)*a(n-1) +8*(6*n-5)*(6*n-1)*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jan 11 2025

A120279 a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}].

Original entry on oeis.org

2, 11, 45, 170, 631, 2346, 8780, 33089, 125466, 478181, 1830258, 7030557, 27088856, 104647615, 405187809, 1571990918, 6109558567, 23782190466, 92705454875, 361834392094, 1413883873953, 5530599237752, 21654401079301, 84859704298176

Offset: 1

Alexander Adamchuk, Jul 05 2006

p divides a(p-1) and a(p-2) for prime p=5,11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1.
p divides a([(2p-1)/2]) for prime p=5,11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1.
p divides a((p-5)/2) for prime p=17,29,41,53,89,101.. =A040115[n] Primes of form 12n+5. Primes congruent to 5 (mod 12) excluding 5.
p divides a((p-5)/3) for prime p=11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1 excluding 5.
p divides a([(p-3)/3]) for prime p=11,17,23,29,41,47,53,59,71..=A007528[n] Primes of form 6n-1 excluding 5.

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

Cf. A007528, A007528, A040115, A048775, A079309, A079309, A066796.

Table[Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}],{n,1,50}]

a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}]. a(n) = A079309(n+1) - (n+1). a(n) = A066796(n+1)/2 - (n+1).
Recurrence: (n+1)*(3*n-2)*a(n) = 6*(3*n^2-1)*a(n-1) - 3*(9*n^2-n-2)*a(n-2) + 2*(2*n-1)*(3*n+1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+3)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012
a(n) = Sum_{k=1..n} Sum_{i=1..k} C(k+i,i). - Wesley Ivan Hurt, Sep 19 2017

A122485 Values of A083097(k) such that A083097(k) = A083097(k+1) - 1.

Original entry on oeis.org

5, 14, 41, 59, 122, 140, 167, 176, 365, 383, 410, 419, 491, 500, 527, 545, 1094, 1112, 1139, 1148, 1220, 1229, 1256, 1274, 1463, 1472, 1499, 1517, 1580, 1598, 1625, 1634, 3281, 3299, 3326, 3335, 3407, 3416, 3443, 3461, 3650, 3659, 3686, 3704, 3767, 3785, 3812

Offset: 1

Alexander Adamchuk, Sep 15 2006

nonn

A083097(n) = A083095(n) = A083096(n)/6 = A083094(n)/4, where A083096 are the Numbers k such that 3 divides Sum_{j=1..k} C(2*j,j) = A066796(k).
All terms are of the form 9*m + 5 and belong to A017221 with m = {0, 1, 4, 6, 13, 15, 18, 19, 40, 42, ...}.
Corresponding numbers m such that a(m) = A083097(m) are A129771 (evil odd numbers).

			A083097 begins {0, 2, 5, 6, 14, 15, 18, 20, 41, 42, 45, 47, 54, 56, 59, 60, ...}.
So a(1) = 5 because 5 = A083097(3) = A083097(3+1) - 1.
a(2) = 14 because 14 = A083097(5) = A083097(5+1) - 1.

Cf. A000069, A000120, A066796, A083094, A083095, A083096, A083097, A010060, A129771.

a(n) = A083097(A129771(n)).

More terms from R. J. Mathar, Jan 17 2008
More terms from Jinyuan Wang, Jan 22 2022
Edited by Michel Marcus, Jan 22 2022

A147304 a(n) = Sum_{k=1..prime(n)^2-1} binomial(2k,k).

Original entry on oeis.org

28, 17576, 43308802158650, 8610524734277600186228691452, 121374542758943982922417964798154019940274699584207321286055873543631298, 8126392396649531937838689708830356413772063825711016912849229977138431439363305375418692100492504264

Offset: 1

N. J. A. Sloane, Apr 25 2009

nonn

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

Cf. A146977, A066796, A147291.

Table[Sum[Binomial[2k,k],{k,Prime[n]^2-1}],{n,7}] (* Harvey P. Dale, Dec 26 2014 *)

a(n) = sum(k=1, prime(n)^2-1, binomial(2*k,k)); \\ Michel Marcus, Jul 07 2018

A054114 T(2n+1,n), array T as in A054110.

Original entry on oeis.org

1, 8, 28, 98, 350, 1274, 4706, 17576, 66196, 250952, 956384, 3660540, 14061140, 54177740, 209295260, 810375650, 3143981870, 12219117170, 47564380970, 185410909790, 723668784230, 2827767747950, 11061198475550, 43308802158650, 169719408596402, 665637941544506

Offset: 0

Clark Kimberling

nonn

Apart from the initial term, identical to A066796(n+1).

Cf. A054110, A066796.

CoefficientList[Series[(-1+3*x+5*x^2-4*x^3+Sqrt[1-4*x])/(x*(1-x)(1-4*x)), {x,0,30}],x] (* Georg Fischer, Apr 09 2020 *)

G.f.: (-1+3*x+5*x^2-4*x^3+sqrt(1-4*x))/(x*(1-x)*(1-4*x)). - Ralf Stephan, Apr 03 2004; corrected by Georg Fischer, Apr 09 2020

For n>0, a(n) = sum(k=1, n+1, C(2k, k)). - Ralf Stephan, Apr 03 2004

Corrected by Franklin T. Adams-Watters, Oct 25 2006

a(23)-a(25) corrected by Georg Fischer, Apr 09 2020

cp's OEIS Frontend

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

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

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

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A167860 Primes p dividing every A167859(m) from m=(p-1)/2 to m=(p-1).

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A146977 a(n) = Sum_{k=1..prime(n)} binomial(2k,k).

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A066798 a(n) = Sum_{i=1..n} binomial(6*i,3*i).

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A120279 a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}].

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A122485 Values of A083097(k) such that A083097(k) = A083097(k+1) - 1.

A066798 a(n) = Sum_{i=1..n} binomial(6i,3i).