cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A346648 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*k,k) / (5*k + 1).

Original entry on oeis.org

1, 2, 9, 73, 751, 8587, 104425, 1323952, 17303503, 231455104, 3153167249, 43597546197, 610232050453, 8629733401556, 123114479858631, 1769728635257503, 25607523627970183, 372688563309335806, 5451995469296025115, 80122698147986922194, 1182341393088427774071
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2021

Keywords

Comments

Binomial transform of A002295.

Links

Crossrefs

Cf. A002295, A007317, A188687, A226910, A346646, A346647, A346649, A346650.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^4 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 20; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/6, 1/3, 1/2, 2/3, 5/6, -n}, {2/5, 3/5, 4/5, 1, 6/5}, -46656/3125], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(6*k,k)/(5*k + 1)); \\ Michel Marcus, Jul 26 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^4 * A(x)^6.
G.f.: Sum_{k>=0} ( binomial(6*k,k) / (5*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 49781^(n + 3/2) / (3359232 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A226974 a(n) = Sum_{k=0..floor(n/3)} binomial(n,3*k)*binomial(4*k,k)/(3*k+1).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 25, 64, 169, 443, 1181, 3224, 8897, 24701, 69161, 195255, 554577, 1583109, 4541461, 13086574, 37856437, 109892403, 320034309, 934774902, 2737689189, 8037746691, 23652564261, 69749727716, 206091735797, 610061655665, 1808962146529
Offset: 0

Views

Author

Karol A. Penson, Jun 25 2013

Keywords

Links

Crossrefs

Cf. A049130, A212385, A226910, A227035.

Programs

  • Maple
    A226974 := proc(n)
    hypergeom([-n/3,-n/3+2/3,-n/3+1/3,1/4,1/2,3/4],[1/3,2/3,2/3,1,4/3],-256/27) ;
    simplify(%) ;
end proc:
seq(A226974(n),n=0..40) ; # R. J. Mathar, Jan 10 2023
  • Mathematica
    Table[Sum[Binomial[n,3*k]*Binomial[4*k,k]/(3*k+1), {k,0,Floor[n/3]}],{n,0,20}] (* Vaclav Kotesovec, Jun 28 2013 *)
  • Maxima
    a(n):=if n<0 then 0 else if n=0 then 1 else sum(sum(sum(a(l)*a(i)*a(j)*a(n-i-j-l-3),l,0,n-3-i-j),j,0,n-3-i),i,0,n-3)+1; /* Vladimir Kruchinin, May 17 2020 */
  • PARI
    a(n) = sum(k=0, n\3, binomial(n,3*k)*binomial(4*k,k)/(3*k+1)); \\ Michel Marcus, Sep 16 2021

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n,3*k)*A002293(k).
Representation in terms of special values of hypergeometric function of type 6F5: a(n) = hypergeom([1/4, 1/2, 3/4, -(1/3)*n, -(1/3)*n+2/3, -(1/3)*n+1/3], [1/3, 2/3, 2/3, 1, 4/3],-4^4/3^3), n>=0.
Recurrence: 27*n*(n+1)*(n-1)*a(n) = 162*n*(n-1)^2*a(n-1) - 81*(5*n^2-15*n+12)*(n-1)*a(n-2) + 4*(199*n^3 - 1098*n^2 + 2043*n - 1296)*a(n-3) - (n-3)*(1173*n^2 - 5097*n + 5584)*a(n-4) + 6*(n-4)*(n-3)*(155*n-401)*a(n-5) - 283*(n-5)*(n-4)*(n-3)*a(n-6). - Vaclav Kotesovec, Jun 28 2013
a(n) ~ (3+4^(1+1/3))^(n+3/2)/(8*3^(n+1)*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 28 2013
G.f. satisfies A(x)=1+x^3*A(x)^4+x/(1-x). - Vladimir Kruchinin, May 17 2020
From Peter Bala, Sep 15 2021: (Start)
O.g.f.: A(x) = (1/x)*series reversion( x*(1 - x^3)/(1 + x*(1 - x^3)) ).
The g.f. of the m-th binomial transform of this sequence is equal to (1/x)*series reversion( x*(1 - x^3)/(1 + (m + 1)*x*(1 - x^3)) ). The case m = -1 gives the sequence [1,0,0,1,0,0,4,0,0,22,0,0,140,...] - an aerated version of A002293. (End)

A227035 a(n) = Sum_{k=0..floor(n/4)} binomial(n,4*k)*binomial(5*k,k)/(4*k+1).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 76, 172, 436, 1156, 3006, 7606, 19202, 49466, 130156, 345356, 915196, 2421532, 6427001, 17163581, 46087911, 124133531, 334850208, 904691576, 2449891276, 6651540676, 18100561856, 49344295152, 134719523056, 368350942416, 1008680051756
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 28 2013

Keywords

Comments

Generally, Sum(binomial(n,p*k)*binomial((p+1)*k,k)/(p*k+1), k=0..floor(n/p)) is asymptotic to (p+(p+1)^(1+1/p))^(n+3/2)/(p^(n+1)*(p+1)^(1+3/(2*p))*n^(3/2)*sqrt(2*Pi)).

Links

Crossrefs

Cf. A002294, A007317 (p=1), A049130 (p=2), A226974 (p=3), A226910 (p=5).

Programs

  • Mathematica
    Table[Sum[Binomial[n,4*k]*Binomial[5*k,k]/(4*k+1),{k,0,Floor[n/4]}],{n,0,20}]
  • PARI
    a(n)=sum(k=0,n\4,binomial(n,4*k)*binomial(5*k,k)/(4*k+1)) \\ Charles R Greathouse IV, Jun 28 2013

Formula

Recurrence: -2869*(n-7)*(n-6)*(n-5)*(n-4)*a(n-8) + 2*(n-6)*(n-5)*(n-4)*(5226*n-17267)*a(n-7) - (n-5)*(n-4)*(11582*n^2-55156*n+50139)*a(n-6) - 3*(n-4)*(612*n^3 - 18926*n^2 + 102684*n - 155665)*a(n-5) + 5*(n-4)*(2959*n^3 - 26172*n^2 + 77408*n - 76800)*a(n-4) - 1024*(n-2)*(2*n-5)*(7*n^2-35*n+48)*a(n-3) + 1024*(n-2)*(n-1)*(7*n^2-28*n+30)*a(n-2) - 1024*(n-2)*(n-1)*n*(2*n-3)*a(n-1) + 256*(n-2)*(n-1)*n*(n+1)*a(n) = 0.
a(n) ~ (4+5^(1+1/4))^(n+3/2)/(4^(n+1)*5^(1+3/8)*n^(3/2)*sqrt(2*Pi)).
G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^4 * A(x)^5. - Ilya Gutkovskiy, Jul 25 2021
From Peter Bala, Sep 15 2021: (Start)
O.g.f.: A(x) = (1/x)*series reversion ( x*(1 - x^4)/(1 + x*(1 - x^4) )).
The g.f. of the m-th binomial transform of this sequence is equal to (1/x)*series reversion ( x*(1 - x^4)/(1 + (m + 1)*x*(1 - x^4)) ). The case m = -1 gives the sequence [1,0,0,0,1,0,0,0,5,0,0,0,35,0,0,0,285,...] - an aerated version of A002294. (End)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.