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 1-6 of 6 results.

A081197 Diagonal sums of A081130.

Original entry on oeis.org

0, 0, 0, 1, 4, 13, 44, 162, 643, 2724, 12259, 58423, 293902, 1555743, 8640526, 50222288, 304792741, 1927313470, 12673784445, 86517541197, 612134881624, 4482215342305, 33919417267456, 264951302794510, 2133720505175351
Offset: 0

Views

Author

Paul Barry, Mar 11 2003

Keywords

Links

Crossrefs

Cf. A081130.

Programs

  • Magma
    [n lt 3 select 0 else (&+[j^(n-j-2)*Binomial(n-j,2): j in [1..n-2]]): n in [0..30]]; // G. C. Greubel, May 15 2021
  • Maple
    A081197 := proc(n)
    add(k^(n-k-2)*binomial(n-k,2), k=1..n-2) ;
end proc: # R. J. Mathar, Feb 13 2015
  • Mathematica
    Table[Sum[k^(n-k-2)*Binomial[n-k, 2], {k,n-2}], {n,0,30}] (* G. C. Greubel, May 15 2021 *)
  • Sage
    [sum( (n-k)^(k-2)*binomial(k,2) for k in (0..n-1) ) for n in (0..30)] # G. C. Greubel, May 15 2021

Formula

a(n) = Sum_{k=1..n-2} k^(n-k-2)*binomial(n-k, 2).
a(n) = Sum_{k=0..n-1} (n-k)^(k-2)*binomial(k, 2). - G. C. Greubel, May 15 2021

Extensions

Terms corrected by G. C. Greubel, May 15 2021

A081131 a(n) = n^(n-2) * binomial(n,2).

Original entry on oeis.org

0, 0, 1, 9, 96, 1250, 19440, 352947, 7340032, 172186884, 4500000000, 129687123005, 4086546038784, 139788510734886, 5159146026151936, 204350482177734375, 8646911284551352320, 389289535005334947848, 18580248257778920521728, 937146152681201173795569
Offset: 0

Views

Author

Paul Barry, Mar 08 2003

Keywords

Comments

Main diagonal of A081130.
a(n) is the number of partial functions f: {1,2,...,n} -> {1,2,...,n} that have exactly 2 undefined elements. - Geoffrey Critzer, Feb 08 2012
a(n+1) is the determinant of the circulant matrix having (n-1, n-2, ..., 0) as first row, for n >= 1. See A070896 for a variant, and A303260 for a related sequence. - M. F. Hasler, Apr 23 2018
a(n) is the number of birooted labeled trees on n nodes. - Brendan McKay, May 01 2018

Links

Crossrefs

Cf. A070896, A081129, A081130, A303260.
Sequences of the form (n+m)^n*binomial(n+2,2): A081133 (m=0), A081132 (m=1), this sequence (m=2), A053507 (m=3), A081196 (m=4).

Programs

  • Magma
    [n lt 2 select 0 else n^(n-2)*Binomial(n,2): n in [0..20]]; // G. C. Greubel, May 18 2021
  • Mathematica
    Join[{0},Table[n^(n-2) Binomial[n, 2], {n, 1, 20}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
  • Sage
    [0 if (n<2) else n^(n-2)*binomial(n,2) for n in (0..20)] # G. C. Greubel, May 18 2021

Formula

a(0) = a(1) = 0, a(n) = n^(n-2)*binomial(n,2).
E.g.f.: T(x)^2/(2*(1-T(x))) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Feb 08 2012

A053507 a(n) = binomial(n-1,2)*n^(n-3).

Original entry on oeis.org

0, 0, 1, 12, 150, 2160, 36015, 688128, 14880348, 360000000, 9646149645, 283787919360, 9098660462034, 315866083233792, 11806916748046875, 472877960873902080, 20205339187128111480, 917543123840934346752, 44131536275846038655193
Offset: 1

Views

Author

N. J. A. Sloane, Jan 15 2000

Keywords

Comments

Number of connected unicyclic simple graphs on n labeled nodes such that the unique cycle has length 3. - Len Smiley, Nov 27 2001
Each simple graph (of this type) corresponds to exactly two 'functional digraphs' counted by A065513.

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Prop. 5.3.2.

Links

Crossrefs

Cf. A000169, A053506, A053508, A053509, A081133, A081132.
Equals 2*A065513. A diagonal of A081130.

Programs

  • GAP
    List([1..20], n-> Binomial(n-1,2)*n^(n-3)); # G. C. Greubel, May 15 2019
  • Magma
    [Binomial(n-1,2)*n^(n-3):n in [1..20]]; // Vincenzo Librandi, Sep 22 2011
  • Magma
    [Binomial(n-1,2)*n^(n-3): n in [1..20]]; // G. C. Greubel, May 15 2019
  • Mathematica
    nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Rest[Range[0, nn]! CoefficientList[Series[t^3/3!, {x, 0, nn}], x]] (* Geoffrey Critzer, Jan 22 2012 *)
Table[Binomial[n-1,2]n^(n-3),{n,20}] (* Harvey P. Dale, Sep 24 2019 *)
  • PARI
    vector(20, n, binomial(n-1,2)*n^(n-3)) \\ G. C. Greubel, Jan 18 2017
  • Sage
    [binomial(n-1,2)*n^(n-3) for n in (1..20)] # G. C. Greubel, May 15 2019

Formula

E.g.f.: -LambertW(-x)^3/3!. - Vladeta Jovovic, Apr 07 2001

A081132 a(n) = (n+1)^n*binomial(n+2,2).

Original entry on oeis.org

1, 6, 54, 640, 9375, 163296, 3294172, 75497472, 1937102445, 55000000000, 1711870023666, 57954652913664, 2120125746145771, 83340051191685120, 3503151123046875000, 156797324626531188736, 7445162356977030877593
Offset: 0

Views

Author

Paul Barry, Mar 08 2003

Keywords

Comments

A diagonal of A081130.
a(n) is the sum of all the fixed points in the set of endofunctions on {1,2,...,n+1}, i.e., the functions f:{1,2,...,n+1} -> {1,2,...,n+1}. - Geoffrey Critzer, Sep 17 2011

Examples

			a(1) = 6 because there are four functions from {1,2} into {1,2}: (1*,1) (1*,2*) (2,1) (2,2*) and the fixed points (marked *) sum to 6.

Links

Crossrefs

Sequences of the form (n+m)^n*binomial(n+2,2): A081133 (m=0), this sequence (m=1), A081131 (m=2), A053507 (m=3), A081196 (m=4).

Programs

  • Magma
    [((n+1)^n*Binomial(n+2,2)): n in [0..20]]; // Vincenzo Librandi, Sep 21 2011
  • Maple
    seq((n+1)^n*binomial(n+2,2), n=0..20); # G. C. Greubel, May 18 2021
  • Mathematica
    Table[n^n*(n+1)/2,{n,20}]
  • Sage
    [(n+1)^n*binomial(n+2,2) for n in (0..20)] # G. C. Greubel, May 18 2021

Formula

a(n) = (n+1)^n*binomial(n+2,2).

A081133 a(n) = n^n*binomial(n+2, 2).

Original entry on oeis.org

1, 3, 24, 270, 3840, 65625, 1306368, 29647548, 754974720, 21308126895, 660000000000, 22254310307658, 811365140791296, 31801886192186565, 1333440819066961920, 59553569091796875000, 2822351843277561397248
Offset: 0

Views

Author

Paul Barry, Mar 08 2003

Keywords

Comments

A diagonal of A081130.

Links

Crossrefs

Sequences of the form (n+m)^n*binomial(n+2,2): this sequence (m=0), A081132 (m=1), A081131 (m=2), A053507 (m=3), A081196 (m=4).

Programs

  • Magma
    [(n^n*Binomial(n+2,2)): n in [0..20]]; // Vincenzo Librandi, Sep 22 2011
  • Maple
    seq(n^n*binomial(n+2,2), n=0..20); # G. C. Greubel, May 18 2021
  • Mathematica
    Join[{1},Table[n^n Binomial[n+2,2],{n,20}]] (* Harvey P. Dale, Dec 27 2011 *)
  • Sage
    [n^n*binomial(n+2,2) for n in (0..20)] # G. C. Greubel, May 18 2021

Formula

a(n) = n^n*(n+1)*(n+2)/2.

A081196 a(n) = (n+4)^n*binomial(n+2,2).

Original entry on oeis.org

1, 15, 216, 3430, 61440, 1240029, 28000000, 701538156, 19349176320, 583247465515, 19090807228416, 674680957031250, 25614222880669696, 1039980693455123385, 44977604109849722880, 2064633276062972568664
Offset: 0

Views

Author

Paul Barry, Mar 11 2003

Keywords

Comments

Diagonal of A081130.

Links

Crossrefs

Sequences of the form (n+m)^n*binomial(n+2,2): A081133 (m=0), A081132 (m=1), A081131 (m=2), A053507 (m=3), this sequence (m=4).

Programs

  • Magma
    [(n+4)^n*Binomial(n+2,2): n in [0..20]]; // Vincenzo Librandi, Aug 07 2013
  • Maple
    seq((n+4)^n*binomial(n+2,2), n=0..20); # G. C. Greubel, May 18 2021
  • Mathematica
    Table[(n+4)^n Binomial[n+2, 2], {n, 0, 30}] (* Vincenzo Librandi, Aug 07 2013 *)
  • Sage
    [(n+4)^n*binomial(n+2,2) for n in (0..20)] # G. C. Greubel, May 18 2021

Formula

a(n) = (n+4)^n*binomial(n+2,2).
Showing 1-6 of 6 results.

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

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