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 1-5 of 5 results.

A052760 Expansion of e.g.f.: x^2*(exp(x)-1)^2.

Original entry on oeis.org

0, 0, 0, 0, 24, 120, 420, 1260, 3472, 9072, 22860, 56100, 134904, 319176, 745108, 1719900, 3931680, 8912352, 20053404, 44825940, 99613960, 220200120, 484441188, 1061157900, 2315254704, 5033163600, 10905189100, 23555209860, 50734299672, 108984793512
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Original name: a simple grammar.

Links

Crossrefs

Cf. A052749.

Programs

  • Magma
    [0,0,0] cat [n*(n-1)*(2^n-8)/4: n in [3..30]]; // Vincenzo Librandi, May 05 2013
  • Maple
    spec := [S,{B=Set(Z,1 <= card),S=Prod(B,B,Z,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    Part[#, Range[1, Length[#], 1]]&@(Array[#!&, Length[#], 0] #)&@CoefficientList[Series[x^2 Exp[x]^2 - 2 Exp[x] x^2 + x^2, {x, 0, 30}], x]//ExpandAll (* Vincenzo Librandi, May 05 2013 *)
  • PARI
    a(n) = if(n<4, 0, n*(n-1)*(2^n-8)/4); \\ Joerg Arndt, May 06 2013

Formula

E.g.f.: x^2*exp(x)^2-2*exp(x)*x^2+x^2.
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (2*n^2+6*n+4)*a(n)+(6-3*n^2-3*n)*a(n+1)+(n^2-n)*a(n+2)}.
For n>=3, a(n) = n*(n-1)*(2^n-8)/4. - Vaclav Kotesovec, Nov 27 2012
a(n) = n*A052749(n-1) = 2*n*(n-1)*Stirling2(n-2,2) for n >= 2. - Andrew Howroyd, Aug 08 2020

Extensions

More terms from Vincenzo Librandi, May 05 2013
Name changed by Andrew Howroyd, Aug 08 2020

A052761 a(n) = 3!*n*S(n-1,3), where S denotes the Stirling numbers of second kind.

Original entry on oeis.org

0, 0, 0, 0, 24, 180, 900, 3780, 14448, 52164, 181500, 615780, 2052072, 6749028, 21976500, 71007300, 228009696, 728451972, 2317445100, 7346047140, 23213772120, 73156412196, 229989358500, 721474964100, 2258832312144, 7059480120900, 22026886599900
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. A000392, A001117, A052749.

Programs

  • Maple
    spec := [S,{B=Set(Z,1 <= card),S=Prod(B,B,B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    Join[{0},Table[3!*n*StirlingS2[n-1,3],{n,30}]] (* Harvey P. Dale, Feb 07 2015 *)
  • PARI
    a(n)={if(n>=1, 3!*n*stirling(n-1, 3, 2), 0)} \\ Andrew Howroyd, Aug 08 2020

Formula

E.g.f.: exp(x)^3*x - 3*exp(x)^2*x + 3*x*exp(x) - x.
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-36*n^2 - 66*n - 6*n^3 - 36)*a(n) + (11*n^3 + 55*n^2 + 66*n)*a(n+1) + (-6*n^3 - 24*n^2 - 18*n)*a(n+2) + (n^3 + 3*n^2 + 2*n)*a(n+3)}
For n>=2, a(n) = n*(3^(n-1) - 3*2^(n-1) + 3). - Vaclav Kotesovec, Nov 27 2012
O.g.f.: 12*x^4*(2 - 9*x + 11*x^2 - 3*x^3)/((1 - 3*x)^2*(1 - 2*x)^2*(1 - x)^2). - Matthew House, Feb 16 2017 [Corrected by Georg Fischer, May 19 2019]
From Andrew Howroyd, Aug 08 2020: (Start)
a(n) = n*A001117(n-1) for n > 1.
E.g.f.: x*(exp(x) - 1)^3. (End)

Extensions

Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001
More terms from Harvey P. Dale, Feb 07 2015

A052776 a(n) = 4!*n*Stirling2(n-1,4).

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 1440, 10920, 67200, 367416, 1864800, 9003720, 41976000, 190788312, 850930080, 3741028200, 16264684800, 70093951608, 299953709280, 1276255985160, 5404640136000, 22796986414104, 95836788986400, 401733800351400, 1679813323056000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. A000919, A052749.

Programs

  • Maple
    spec := [S,{B=Set(Z,1 <= card),S=Prod(Z,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    Join[{0},Table[4!*n*StirlingS2[n-1,4],{n,30}]] (* Harvey P. Dale, Dec 24 2014 *)
  • PARI
    a(n)={if(n>=1, 4!*n*stirling(n-1, 4, 2), 0)} \\ Andrew Howroyd, Aug 08 2020

Formula

E.g.f.: x*exp(x)^4-4*exp(x)^3*x+6*exp(x)^2*x-4*x*exp(x)+x.
Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (1200*n+840*n^2+240*n^3+576+24*n^4)*a(n)+(-1200*n-1300*n^2-450*n^3-50*n^4)*a(n+1)+(35*n^4+420*n+665*n^2+280*n^3)*a(n+2)+(-80*n-10*n^4-140*n^2-70*n^3)*a(n+3)+(n^4+6*n^3+11*n^2+6*n)*a(n+4), a(5)=120}.
From Andrew Howroyd, Aug 08 2020: (Start)
a(n) = n*A000919(n-1) for n > 1.
G.f.: 24*x^5*(5 - 40*x + 105*x^2 - 100*x^3 + 24*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x))^2.
E.g.f.: x*(exp(x) - 1)^4. (End)

Extensions

Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001

A363603 Expansion of e.g.f. (1/4)*(exp(x)-x-1)*(exp(x)-1)^2.

Original entry on oeis.org

3, 20, 90, 343, 1197, 3966, 12720, 39941, 123651, 379132, 1154790, 3501219, 10581465, 31908218, 96068700, 288926977, 868288239, 2608010424, 7830584850, 23505386015, 70544469573, 211692128950, 635198021640, 1905845723133, 5718057263067
Offset: 4

Views

Author

Enrique Navarrete, Jun 11 2023

Keywords

Comments

4*a(n) is the number of ordered set partitions of an n-set into 3 nonempty sets such that the number of elements in a particular set (say the first one) is at least two (see example).
4*a(n) is also the number of ternary strings using digits {0,1,2} so that all digits are used and a particular digit appears at least twice; for example, for n=5, the 80 strings with at least two 0's are 00112 (30 of this type), 00122 (30 of this type), 00012 (20 of this type).

Examples

			4*a(5)=80 since the ordered set partitions are the following: 30 of type {1,2}{3,4},{5}; 30 of type {1,2},{3},{4,5}; 20 of type {1,2,3},{4},{5}.

Links

Crossrefs

Cf. A000225, A052749, A001117, A363591.

Programs

  • Mathematica
    A363603[n_]:=(3^n-3(2^n-1))/4-(n/2)(2^(n-2)-1);Array[A363603,40,4] (* or *)
LinearRecurrence[{9,-31,51,-40,12},{3,20,90,343,1197},40] (* Paolo Xausa, Nov 18 2023 *)

Formula

a(n) = (3^n - 3*(2^n - 1))/4 - (n/2)*(2^(n-2) - 1), n>=4.
G.f.: x^4*(3 - 7*x + 3*x^2)/((1 - 3*x)*(1 - 2*x)^2*(1 - x)^2). - Stefano Spezia, Jun 11 2023
a(n) = (Sum_{k=2..n-2} A000225(k-1)*binomial(n,k))/2. - R. J. Cano, Jul 27 2023

A052785 a(n) = 5!*n*Stirling2(n-1, 5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 12600, 134400, 1134000, 8341200, 56133000, 355291200, 2151864000, 12614281680, 72135063000, 404672486400, 2236228722000, 12209943566160, 66024457842600, 354214283304000, 1887999348060000, 10008933180578640, 52820388477271800, 277680637970208000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Original name: a simple grammar.

Links

Crossrefs

Cf. A001118, A052749.

Programs

  • Maple
    spec := [S,{B=Set(Z,1 <= card),S=Prod(Z,B,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • PARI
    a(n)={if(n>=1, 5!*n*stirling(n-1, 5, 2), 0)} \\ Andrew Howroyd, Aug 08 2020

Formula

E.g.f.: x*exp(x)^5-5*x*exp(x)^4+10*exp(x)^3*x-10*exp(x)^2*x+5*x*exp(x)-x.
Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, ( - 14400 - 27000*n^2 - 32880*n - 120*n^5 - 1800*n^4 - 10200*n^3)*a(n) + (42196*n^2 + 19454*n^3 + 32880*n + 274*n^5 + 3836*n^4)*a(n + 1) + ( - 13500*n - 13275*n^3 - 24075*n^2 - 225*n^5 - 2925*n^4)*a(n + 2) + (85*n^5 + 3400*n + 1020*n^4 + 4165*n^3 + 6630*n^2)*a(n + 3) + ( - 915*n^2 - 450*n - 615*n^3 - 15*n^5 - 165*n^4)*a(n + 4) + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a(n + 5)}.
From Andrew Howroyd, Aug 08 2020: (Start)
a(n) = n*A001118(n-1) for n > 1.
G.f.: 120*x^6*(2 - 5*x)*(3 - 30*x + 95*x^2 - 100*x^3 + 24*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x))^2.
E.g.f.: x*(exp(x) - 1)^5. (End)

Extensions

Name changed and terms a(21) and beyond from Andrew Howroyd, Aug 08 2020
Showing 1-5 of 5 results.

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