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.

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A098559 Expansion of e.g.f. (1+3*x)/(1-3*x).

Original entry on oeis.org

1, 6, 36, 324, 3888, 58320, 1049760, 22044960, 529079040, 14285134080, 428554022400, 14142282739200, 509122178611200, 19855764965836800, 833942128565145600, 37527395785431552000, 1801314997700714496000, 91867064882736439296000, 4960821503667767721984000
Offset: 0

Views

Author

Paul Barry, Sep 14 2004

Keywords

Links

Crossrefs

Cf. A098558, A032184.

Programs

  • Magma
    [2*3^n*Factorial(n)-0^n: n in [0..20]]; // Vincenzo Librandi, Aug 08 2014
  • Mathematica
    Join[{1},Table[2*3^n n!,{n,20}]] (* Harvey P. Dale, Aug 08 2014 *)

Formula

a(n) = 6 * A034001(n) for n >= 1.
a(n) = 2*3^n*n! - 0^n.
Sum_{n>=0} 1/a(n) = (exp(1/3)+1)/2. - Amiram Eldar, Feb 02 2023

A081406 a(n) = (n+1)*a(n-3), a(0)=a(1)=a(2)=1 for n>1.

Original entry on oeis.org

1, 1, 1, 4, 5, 6, 28, 40, 54, 280, 440, 648, 3640, 6160, 9720, 58240, 104720, 174960, 1106560, 2094400, 3674160, 24344320, 48171200, 88179840, 608608000, 1252451200, 2380855680, 17041024000, 36321084800, 71425670400, 528271744000, 1162274713600
Offset: 0

Views

Author

Labos Elemer, Apr 01 2003

Keywords

Examples

			a(3n+2)=A034001[n]; while other subsequences are near(but not equal) to A001669, A000359.

Links

Crossrefs

Cf. A002866, A000142, A001147, A081405.

Programs

  • GAP
    a:= function(k)
    if k<3 then return 1;
    elif k<6 then return k+1;
    else return (k+1)*a(k-3);
    fi;
  end;
List([0..35], n-> a(n) ); # G. C. Greubel, Aug 24 2019
  • Magma
    a:= func< n | n le 2 select 1 else n in [3..5] select n+1 else (n+1)*Self(n-2) >;
[a(n): n in [0..35]]; // G. C. Greubel, Aug 24 2019
  • Mathematica
    f[n_]:= (n+1)*f[n-3]; f[0]=1; f[1]=1; f[2]=1; Table[f[n], {n, 30}]
RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==(n+1)a[n-3]},a,{n,30}] (* Harvey P. Dale, Mar 06 2019 *)
  • PARI
    a(n) = if(n<3, 1, (n+1)*a(n-3) );
vector(35, n, a(n-1)) \\ G. C. Greubel, Aug 24 2019
  • Sage
    def a(n):
    if n<3: return 1
    elif 3<= n <= 5: return n+1
    else: return (n+1)*a(n-3)
[a(n) for n in (0..35)] # G. C. Greubel, Aug 24 2019

Extensions

Corrected and extended by Harvey P. Dale, Mar 06 2019

A081408 a(n) = (n+1)*a(n-5), with a(0)=a(1)=a(2)=a(3)=a(4)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 7, 8, 9, 10, 66, 84, 104, 126, 150, 1056, 1428, 1872, 2394, 3000, 22176, 31416, 43056, 57456, 75000, 576576, 848232, 1205568, 1666224, 2250000, 17873856, 27143424, 39783744, 56651616, 78750000, 643458816, 1004306688, 1511782272
Offset: 0

Views

Author

Labos Elemer, Apr 01 2003

Keywords

Comments

Quintic factorial sequences are generated by single 5-order recursion and appear in unified form.

Examples

			A008548, A034323, A034300, A034301, A034325 sequences are combed together as A081408(5n+r) with r=0,1,2,3,4.

Links

Crossrefs

Cf. A001147, A002866, A034001, A007599, A034000, A007696, A000407, A034176, A034177, A008548, A034323, A034300, A034301, A034325 [double, triple, quartic, quintic, factorial subsequences], generated together in A081405-A081408.

Programs

  • GAP
    a:=[1,1,1,1,1];; for n in [6..40] do a[n]:=n*a[n-5]; od; a; # G. C. Greubel, Aug 15 2019
  • Haskell
    a081407 n = a081408_list !! n
a081407_list = 1 : 1 : 1 : 1 : zipWith (*) [5..] a081407_list
-- Reinhard Zumkeller, Jan 05 2012
  • Magma
    [n le 5 select 1 else n*Self(n-5): n in [1..40]]; // G. C. Greubel, Aug 15 2019
  • Mathematica
    a[0]=a[1]=a[2]=a[3]=a[4]=1; a[x_]:= (x+1)*a[x-5]; Table[a[n], {n, 40}]
  • PARI
    m=30; v=concat([1,1,1,1,1], vector(m-5)); for(n=6, m, v[n]=n*v[n-5] ); v \\ G. C. Greubel, Aug 15 2019
  • Sage
    def a(n):
    if (n<5): return 1
    else: return (n+1)*a(n-5)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 15 2019

A368028 Square array read by antidiagonals; T(n,k) = number of ways a vehicle with capacity k can transport n distinct individuals with distinct starting and finishing points.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 6, 6, 1, 1, 0, 24, 54, 6, 1, 1, 0, 120, 648, 90, 6, 1, 1, 0, 720, 9720, 1944, 90, 6, 1, 1, 0, 5040, 174960, 52920, 2520, 90, 6, 1, 1, 0, 40320, 3674160, 1730160, 99000, 2520, 90, 6, 1, 1, 0, 362880, 88179840, 65998800, 4806000, 113400, 2520, 90, 6, 1, 1, 0, 3628800, 2380855680, 2877275520, 274050000, 6966000, 113400, 2520, 90, 6, 1, 1
Offset: 0

Views

Author

Henry Bottomley, Dec 24 2023

Keywords

Examples

			T(3,2)=54 represented by the nine patterns AABBCC, AABCBC, AABCCB, ABABCC, ABACBC, ABACCB, ABBACC, ABBCAC, ABBCCA multiplied by 3!=6 for the permutations of A,B,C; but for example ABCABC would not work as the vehicle would be over its capacity of 2 after picking up 3 passengers.

Links

Crossrefs

Cf. A080934. Rows include A000012, A057427. Columns include A000007, A000142, A034001. Diagonals include A000680 and A071798.

Formula

If f(n,k,c)=n*f(n-1,k,c+1)+c*f(n,k,c-1) with f(n,k,c)=0 when n<0 or k<0 or c<0 or k
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