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.

A002058 Number of internal triangles in all triangulations of an (n+1)-gon.

Original entry on oeis.org

2, 14, 72, 330, 1430, 6006, 24752, 100776, 406980, 1634380, 6537520, 26075790, 103791870, 412506150, 1637618400, 6495886320, 25751549340, 102042235620, 404225281200, 1600944863700, 6339741660252, 25103519174844, 99399793096352
Offset: 5

Views

Author

N. J. A. Sloane

Keywords

Comments

From Richard Stanley, Jan 30 2014: (Start)
The previous name "Number of partitions of a n-gon into (n-3) parts" was erroneous.
Cayley does not seem to have a combinatorial interpretation of this sequence. He just uses it as an auxiliary sequence, nor am I aware of a combinatorial interpretation in the literature.
(End)
First subdiagonal of the table of V(r,k) on page 240. The values V(11,8) = 24052, V(13,10)= 396800 and V(15,12)= 6547520 of the publication are replaced/corrected in the sequence.

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002059, A002060.

Programs

  • PARI
    x='x+O('x^66); Vec(64*x^5/((1+sqrt(1-4*x))^5*sqrt(1-4*x))) \\ Joerg Arndt, Jan 30 2014

Formula

a(n) = 2*binomial(2*n-5,n-5) = 2*A003516(n-3). - David Callan, Mar 30 2007
G.f. 64*x^5/((1+sqrt(1-4*x))^5*sqrt(1-4*x)). - R. J. Mathar, Nov 27 2011
a(n) ~ 4^n/(16*sqrt(Pi*n)). - Ilya Gutkovskiy, Apr 11 2017

Extensions

Definition corrected by Richard Stanley, Jan 30 2014

A002060 Number of partitions of an n-gon into (n-5) parts.

Original entry on oeis.org

4, 60, 550, 4004, 25480, 148512, 813960, 4263600, 21573816, 106234700, 511801290, 2421810300, 11289642000, 51967090560, 236635858800, 1067518772640, 4776759725400, 21221827263000, 93687293423724, 411270420524040, 1796296260955504, 7809983743284800, 33816739954270000
Offset: 7

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) = V(r=n,k=n-5), 4th subdiagonal of the triangle of V on page 240.
It appears that V(r=15,k=10) in the Cayley table is an error, so the sequence was intended to be 4, 60, 550, 4004, 25480, 148512, 813960, 4263600, 21573816, 106234700, 511801290, 2421810300, 11289642000, 51967090560, 236635858800... - R. J. Mathar, Nov 26 2011

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.

Crossrefs

Cf. A002058, A002059.

Programs

  • Maple
    V := proc(r,k)
        local a ,t;
        a := k-1;
        for t from k-2 to 1 by -1 do
                a := a*(r+t)/(t+2) ;
        end do:
        for t from 3 to k+1 do
                a := a*(r-t)/(k-t+2) ;
        end do:
        a ;
end proc:
A002060 := proc(n)
        V(n,n-5) ;
end proc:
seq(A002060(n),n=7..25) ; # R. J. Mathar, Nov 26 2011
  • Mathematica
    V[r_, k_] := Module[{a, t}, a = k - 1;
   For[t = k - 2, t >= 1, t--, a = a*(r + t)/(t + 2)];
   For[t = 3, t <= k + 1, t++, a = a*(r - t)/(k - t + 2)];
   a];
A002060[n_] := V[n, n - 5];
Table[A002060[n], {n, 7, 29}] (* Jean-François Alcover, Mar 10 2023, after R. J. Mathar *)

Extensions

More terms from Hugo Pfoertner, Dec 26 2021

A259476 Cayley's triangle of V numbers; triangle V(n,k), n >= 4, n <= k <= 2*n-4, read by rows.

Original entry on oeis.org

1, 2, 4, 3, 14, 14, 4, 32, 72, 48, 5, 60, 225, 330, 165, 6, 100, 550, 1320, 1430, 572, 7, 154, 1155, 4004, 7007, 6006, 2002, 8, 224, 2184, 10192, 25480, 34944, 24752, 7072, 9, 312, 3822, 22932, 76440, 148512, 167076, 100776, 25194, 10, 420, 6300, 47040, 199920, 514080, 813960, 775200, 406980, 90440, 11, 550, 9900, 89760, 471240, 1534896, 3197700, 4263600, 3517470, 1634380, 326876
Offset: 4

Views

Author

N. J. A. Sloane, Jul 03 2015

Keywords

Examples

			Triangle begins:
  1;
     2, 4;
        3, 14, 14;
            4, 32, 72,  48;
                5, 60, 225, 330,  165;
                    6, 100, 550, 1320, 1430, 572;
  ...

Links

  • A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.

Crossrefs

Diagonals give A002057, A002058, A002059, A002060.
Row sums give A065096 (with a different offset).

Programs

  • Maple
    V := proc(n,x)
    local X,g,i ;
    X := x^2/(1-x) ;
    g := X^n ;
    for i from 1 to n-2 do
        g := diff(g,x) ;
    end do;
    x^2*g*2*(n-1)/n! ;
end proc;
A259476 := proc(n,k)
    V(k-n+2,x) ;
    coeftayl(%,x=0,n+2) ;
end proc:
for n from 4 to 14 do
    for k from n to 2*n-4 do
        printf("%d,",A259476(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 09 2015
  • Mathematica
    T[n_, m_] := 2 Binomial[m, n] Binomial[n-2, m-n+2]/(n-2);
Table[T[n, m], {n, 4, 14}, {m, n, 2n-4}] // Flatten (* Jean-François Alcover, Apr 15 2023, after Vladimir Kruchinin *)
  • Maxima
    T(n,m):=if n<4 then 0 else (2*binomial(m,n)*binomial(n-2,m-n+2))/(n-2); /* Vladimir Kruchinin, Jan 27 2022 */

Formula

G.f.: (1-x*y*(1+2*y)-sqrt(1-2*x*y*(1+2*y)+x^2*y^2))^2/(4*y^4*(1+y)^2). - Vladimir Kruchinin, Jan 27 2022
T(n,m) = 2*C(m,n)*C(n-2,m-n+2)/(n-2), n>=4. - Vladimir Kruchinin, Jan 27 2022
Showing 1-3 of 3 results.

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