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.

A090407 a(n) = Sum_{k = 0..n} C(4*n + 1, 4*k).

Original entry on oeis.org

1, 6, 136, 2016, 32896, 523776, 8390656, 134209536, 2147516416, 34359607296, 549756338176, 8796090925056, 140737496743936, 2251799780130816, 36028797153181696, 576460751766552576, 9223372039002259456
Offset: 0

Views

Author

Paul Barry, Nov 29 2003

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Sum[Binomial[4n+1,4k],{k,0,n}],{n,0,30}] (* or *) LinearRecurrence[ {12,64},{1,6},30] (* Harvey P. Dale, Jan 19 2012 *)

Formula

From Harvey P. Dale, Jan 19 2012: (Start)
a(0) = 1, a(1) = 6, a(n) = 12*a(n-1)+64*a(n-2).
G.f.: (6*x-1)/(64*x^2+12*x-1). (End)
a(n) = (1/2) * 4^n * (4^n + (-1)^n). - Peter Bala, Feb 06 2019

A345457 a(n) = Sum_{k=0..n} binomial(5*n+3,5*k).

Original entry on oeis.org

1, 57, 1574, 53143, 1669801, 53774932, 1717012749, 54986385093, 1759098789526, 56296324109907, 1801425114687749, 57646238657975068, 1844672594930734801, 59029601136140621857, 1888946370232447241574, 60446293452901248074943
Offset: 0

Views

Author

Seiichi Manyama, Jun 20 2021

Keywords

Crossrefs

Sum_{k=0..n} binomial(b*n+c,b*k): A090408 (b=4,c=3), A070782 (b=5,c=0), A345455 (b=5,c=1), A345456 (b=5,c=2), this sequence (b=5,c=3), A345458 (b=5,c=4).
Cf. A139398.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[5*n + 3, 5*k], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Jun 20 2021 *)
  • PARI
    a(n) = sum(k=0, n, binomial(5*n+3, 5*k));
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec((1+36*x+24*x^2)/((1-32*x)*(1+11*x-x^2)))

Formula

G.f.: (1 + 36*x + 24*x^2) / ((1 - 32*x)*(1 + 11*x - x^2)).
a(n) = 21*a(n-1) + 353*a(n-2) - 32*a(n-3) for n>2.
a(n) = A139398(5*n+3).
a(n) = 2^(5*n + 4)/10 + (( 475 - 213*sqrt(5))/phi^(5*n) + ( 65 - 33*sqrt(5))*(-1)^n*phi^(5*n)) / (10*(41*sqrt(5)-90)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 20 2021

A090411 Expansion of g.f. (1-x)/(1-16*x).

Original entry on oeis.org

1, 15, 240, 3840, 61440, 983040, 15728640, 251658240, 4026531840, 64424509440, 1030792151040, 16492674416640, 263882790666240, 4222124650659840, 67553994410557440, 1080863910568919040, 17293822569102704640, 276701161105643274240, 4427218577690292387840
Offset: 0

Views

Author

Paul Barry, Nov 30 2003

Keywords

Crossrefs

Programs

Formula

a(n) = 15*16^(n-1) + 0^n/16.
a(n) = Sum_{j=0..3, Sum_{k=0..n, C(4*n+j, 4*k)}}.
a(n) = (A070775(n) + A090407(n) + A001025(n) + A090408(n))/4.
From Elmo R. Oliveira, Mar 25 2025: (Start)
E.g.f.: (15*exp(16*x) + 1)/16.
a(n) = 16*a(n-1) for n > 1. (End)
Showing 1-3 of 3 results.