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.

A382433 a(n) = S(6,n), where S(r,n) = Sum_{k=0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

Original entry on oeis.org

1, 1, 2, 65, 794, 19722, 562692, 15105729, 553537490, 18107304842, 716747344436, 27247858130506, 1137502720488532, 47573235297987700, 2085487143991309320, 92820152112054862785, 4246321874111740074210, 197525644801830489637170, 9363425291004877645851300
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2025

Keywords

Crossrefs

Column k=6 of A357824.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
         `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
        end:
    a:= n-> add(b(n, n-2*j)^6, j=0..n/2):
    seq(a(n), n=0..18);  # Alois P. Heinz, Mar 25 2025
  • Mathematica
    Table[Sum[Binomial[n,k] * (Binomial[n,k] - Binomial[n,k-1])^5, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 25 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^5);
    
  • Python
    from math import comb
    def A382433(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

Formula

a(n) = Sum_{k=0..floor(n/2)} A008315(n,k)^6.
a(n) = Sum_{k=0..n} A120730(n,k)^6.
a(n) = A357824(n,6).
a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^5.
a(n) ~ 5 * 2^(6*n+4) / (3^(5/2) * Pi^(5/2) * n^(11/2)). - Vaclav Kotesovec, Mar 25 2025

A382434 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^4.

Original entry on oeis.org

1, 1, 3, 33, 195, 1763, 15623, 156257, 1630947, 17911299, 203739015, 2389928995, 28749060871, 353362388551, 4424242664975, 56290517376737, 726355164976547, 9490129871680355, 125375330053632455, 1672895457018337859, 22522481793315373319, 305695116823973096519
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2025

Keywords

Crossrefs

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
         `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
        end:
    a:= n-> 2*add(b(n, n-2*j)^4, j=0..n/2)-1:
    seq(a(n), n=0..21);  # Alois P. Heinz, Mar 25 2025
  • PARI
    a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^4);
    
  • Python
    from math import comb
    def A382434(n): return (sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**4 for j in range((n>>1)+1))<<1)-1 # Chai Wah Wu, Mar 25 2025

Formula

a(n) = Sum_{k=0..n} A080233(n,k)^4 = Sum_{k=0..n} A156644(n,k)^4.
a(n) = 2 * A129123(n) - 1.
D-finite with recurrence n*(n+1)^3*a(n) -2*n*(11*n^3-17*n^2+5*n+5)*a(n-1) -4*(n-1)*(70*n^3-365*n^2+527*n-162)*a(n-2) +8*(n-2)*(584*n^3-5020*n^2+14111*n-13059)*a(n-3) +1344*(4*n-11)*(4*n-13)*(-3+n)^2*a(n-4) +9*(2875*n^4-33975*n^3+149945*n^2-293541*n+215336)=0. - R. J. Mathar, Mar 31 2025

A382446 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^6.

Original entry on oeis.org

1, 1, 4, 257, 4286, 258952, 11816512, 632854273, 43732565914, 2637804065366, 207379028199080, 14568483339859880, 1205457271871693920, 95108827011788280160, 8187664948710535579904, 698818327346476962092801, 62477582066507173352034866, 5627626080883126186936773514
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2025

Keywords

Crossrefs

Programs

  • Magma
    [&+[Binomial(n, k)* (Binomial(n, k) - Binomial (n, k-1))^6: k in [0..n]]: n in [0..21]]; // Vincenzo Librandi, Mar 30 2025
  • Mathematica
    Table[Sum[Binomial[n,k]*(Binomial[n,k]-Binomial[n,k-1])^6,{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^6);
    

Formula

a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^5.
a(n) ~ 15 * 2^(7*n+9) / (Pi^3 * 7^(7/2) * n^6). - Vaclav Kotesovec, Mar 26 2025
Showing 1-3 of 3 results.