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.

Previous Showing 11-16 of 16 results.

A028685 Galois numbers for p=23; order of group AGL(n,23).

Original entry on oeis.org

1, 506, 141331872, 20920469730667584, 1638296742744745305180456960, 67868907839960050279986415163868117749760, 1487321615877089920298398794877451264100990832314711736320
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A028365, A028665, A028667, A028669, A028673, A028675, A028679, A028681.

Programs

  • Mathematica
    FoldList[ #1*23^#2 (23^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 23^n * Product[23^n - 23^k, {k, 0, n-1}]; Array[a, 7, 0] (* Amiram Eldar, Jul 12 2025 *)
  • PARI
    a(n) = 23^n * prod(k = 0, n-1, 23^n - 23^k); \\ Amiram Eldar, Jul 12 2025

Formula

a(n) = 23^n * Product_{k=0..n-1} (23^n - 23^k).
a(n) ~ c * 23^(n^2+n), where c = Product_{k>=1} (1 - 1/23^k) = 0.954631535623... . - Amiram Eldar, Jul 12 2025

A203305 Vandermonde determinant of the first n terms of (1,3,7,15,31,...).

Original entry on oeis.org

1, 2, 48, 64512, 20808990720, 6658450862270054400, 8590449816558320728896700416000, 180165778137909187135292776823951671626301440000, 246665746050863452218796304775365273357060390005370386894553088000000
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2012

Keywords

Comments

Each term divides its successor, as in A028365 and A203307.

Links

Crossrefs

Cf. A000225, A005329, A028365, A203307, A335011.

Programs

  • Magma
    [1] cat [(&*[(&*[2^(k+1) - 2^j: j in [1..k]]): k in [1..n-1]]): n in [2..15]]; // G. C. Greubel, Aug 30 2023
  • Mathematica
    (* First program *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]         (* A203305 *)
Table[v[n+1]/v[n], {n,z}]  (* A028365 *)
%/2                         (* A203307 *)
(* Second program *)
Table[(-1)^n * 2^(n*(n+1)*(2*n+1)/6 - 1) / QPochhammer[2, 2, n] * Product[QPochhammer[1/2^k, 2, k], {k, 2, n}], {n, 10}] (* Vaclav Kotesovec, Feb 18 2021 *)
  • SageMath
    [product(product(2^k - 2^j for j in range(1,k)) for k in range(2,n+1)) for n in range(1,16)] # G. C. Greubel, Aug 30 2023

Formula

a(n) = Product_{k=1..n-1} Product_{j=1..k} (2^(k+1) - 2^j).
From Vaclav Kotesovec, Feb 18 2021: (Start)
a(n) = (-1)^n * (2^(n*(n+1)*(2*n+1)/6 - 1) / QPochhammer(2,2,n)) * Product_{k=2..n} QPochhammer(1/2^k, 2, k).
a(n) ~ 2^(n*(n^2 - 1)/3) * QPochhammer(1/2)^n / A335011. (End)
a(n) = Product_{k=2..n} ( 2^(k+1)^2 * QPochhammer(2^(-k-1), 2, k+1) )/ (2^(k+1) - 1). - G. C. Greubel, Aug 30 2023

A362596 Number of parking functions of size n avoiding the patterns 213 and 321.

Original entry on oeis.org

1, 1, 3, 13, 60, 275, 1238, 5480, 23922, 103267, 441798, 1876366, 7921488, 33275758, 139194812, 580180598, 2410827422, 9990993443, 41308185542, 170439003998, 701953309592, 2886284314298, 11850433719572, 48591008205608, 199002198798980, 814117064956430
Offset: 0

Views

Author

Lara Pudwell, Apr 27 2023

Keywords

Examples

			For n=3 the a(3)=13 parking functions, given in block notation, are {1},{2},{3}; {1,2},{},{3}; {1,2},{3},{}; {1},{2,3},{}; {1,2,3},{},{}; {1},{3},{2}; {1,3},{},{2}; {1,3},{2},{}; {2},{3},{1}; {2,3},{},{1}; {2,3},{1},{}; {3},{1},{2}; {3},{1,2},{}.

Links

Crossrefs

Cf. A000108, A028364, A028365, A362597.

Programs

  • PARI
    a(n)=if(n==0, 1, (n^2 - 3*n + 4)*binomial(2*n,n)/(4*(n+1)) + 4^n/8) \\ Andrew Howroyd, Apr 27 2023
  • Python
    from math import comb
def A362596(n): return ((n*(n-3)+4)*comb(n<<1,n)//(n+1)>>2)+(1<<(n<<1)-3) if n>1 else 1 # Chai Wah Wu, Apr 27 2023

Formula

For n>=1, a(n) = (n^2 - 3*n + 4)/4*A000108(n) + 4^(n - 1)/2.
For n>=1, a(n) = A000108(n) + Sum_{m=1..n-1} m*A028364(n-1,m-1).
G.f.: 1+((9*x^2 - 10*x + 2)*sqrt(1 - 4*x) - 23*x^2 + 14*x - 2)/(2*(1 - 4*x)^(3/2)*x).
D-finite with recurrence 2*(n+1)*a(n) +2*(-15*n+1)*a(n-1) +(167*n-193)*a(n-2) +2*(-204*n+467)*a(n-3) +184*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jan 11 2024

A362597 Number of parking functions of size n avoiding the patterns 213 and 312.

Original entry on oeis.org

1, 1, 3, 12, 54, 259, 1293, 6634, 34716, 184389, 990711, 5372088, 29347794, 161317671, 891313569, 4946324886, 27552980088, 153982124809, 862997075691, 4848839608228, 27304369787694, 154059320699211, 870796075968693, 4929937918315522, 27950989413184404
Offset: 0

Views

Author

Lara Pudwell, Apr 27 2023

Keywords

Examples

			For n=3 the a(3)=12 parking functions, given in block notation, are {1},{2},{3}; {1,2},{},{3}; {1,2},{3},{}; {1},{2,3},{}; {1,2,3},{},{}; {1},{3},{2}; {1,3},{},{2}; {1,3},{2},{}; {2},{3},{1}; {2,3},{},{1}; {2,3},{1},{}; {3},{2},{1}.

Links

Crossrefs

Cf. A028365, A362596.

Programs

  • Maple
    A362597 := proc(n)
    if n = 0 then
        1;
    else
        add(add(binomial(n - 1, i)*(k + 1)*binomial(2*n - 2 - k, n - 1 - k)/n,i=0..k),k=0..n-1) ;
    end if;
end proc:
seq(A362597(n),n=0..60) ; # R. J. Mathar, Jan 11 2024
  • PARI
    a(n)={0^n + sum(k=0, n-1, sum(i=0, k, binomial(n - 1, i)*(k + 1)*binomial(2*n - 2 - k, n - 1 - k)/n))} \\ Andrew Howroyd, Apr 27 2023

Formula

For n>=1, a(n) = Sum_{k=0..n-1} Sum_{i=0..k} binomial(n - 1, i)*(k + 1)*binomial(2*n - 2 - k, n - 1 - k)/n.
D-finite with recurrence (n+1)*a(n) +3*(-4*n+1)*a(n-1) +(34*n-45)*a(n-2) +3*(4*n-17)*a(n-3) +3*(-n+4)*a(n-4)=0. - R. J. Mathar, Jan 11 2024

A203307 a(n) = v(n+1)/(2*v(n)), where v = A203305.

Original entry on oeis.org

1, 12, 672, 161280, 159989760, 645078712320, 10486399547473920, 684552162459097497600, 179100751368498596492083200, 187617350297573441752474740326400, 786539962489104046627462744981792358400
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2012

Keywords

Links

Crossrefs

Cf. A203305, A028365.

Programs

  • Magma
    [(&*[2^(n+1) - 2^(j+1): j in [0..n-1]])/2: n in [1..20]]; // G. C. Greubel, Aug 31 2023
  • Mathematica
    (* First program *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]         (* A203305 *)
Table[v[n+1]/v[n], {n,z}]  (* A028365 *)
%/2                         (* A203307 *)
(* Second program *)
Table[(-1)^n*2^Binomial[n+1,2]*QPochhammer[2,2,n]/2, {n,20}] (* G. C. Greubel, Aug 31 2023 *)
  • SageMath
    [product(2^(n+1) - 2^(k+1) for k in range(n))/2 for n in range(1,21)] # G. C. Greubel, Aug 31 2023

Formula

a(n) = (1/2)*A028365(n) for n>0.
a(n) = (-1)^n * 2^(binomial(n+1,2) - 1) * QPochhammer(2,2,n). - G. C. Greubel, Aug 31 2023

A377642 a(n) = (1/(n-1)!) * Product_{i=1..n-1} (2^n-2^i).

Original entry on oeis.org

1, 2, 12, 224, 13440, 2666496, 1791885312, 4161269661696, 33955960439439360, 987107315743488737280, 103404624282172311371513856, 39408968779516596852827017445376, 55084280201257118417007491904448757760, 284322478318511376197290687371005495020093440
Offset: 1

Views

Author

Nikita Babich, Nov 05 2024

Keywords

Links

Crossrefs

Appears to be main diagonal of A270882.
Cf. A000079, A000142, A002884, A028365, A048651, A053601.

Programs

  • Mathematica
    Table[Product[2^n - 2^i, {i, 1, n - 1}]/Factorial[n - 1], {n, 1, 20}]
  • PARI
    a(n)=prod(i=1, n-1, 2^n-2^i)/(n-1)! \\ Andrew Howroyd, Nov 10 2024

Formula

a(n) = (Product_{i=1..n-1}(2^n-2^i))/((n-1)!).
a(n) = A028365(n-1)/A000142(n-1).
a(n) = A000079(n-1) * A053601(n-1).
a(n) ~ A048651 * 2^(n*(n-1)) / (n-1)!. - Vaclav Kotesovec, Nov 13 2024
Previous Showing 11-16 of 16 results.

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

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