A002060 -id:A002060 - cp's OEIS frontend

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

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

N. J. A. Sloane

nonn

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.

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).

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

Cf. A002059, A002060.

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

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

Definition corrected by Richard Stanley, Jan 30 2014

A002059 Number of partitions of an n-gon into (n-4) parts.

Original entry on oeis.org

3, 32, 225, 1320, 7007, 34944, 167076, 775200, 3517470, 15690048, 69052555, 300638520, 1297398375, 5557977600, 23663585880, 100222246080, 422559514170, 1774647576000, 7427639542050, 30994292561232, 128989359164358

Offset: 6

N. J. A. Sloane

nonn

Second subdiagonal of the table of values of V(r,k) on page 240.

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).

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

Cf. A002058, A002060.

a(n) = (n-3) * binomial(2n-6,n). - Gill Barequet, Nov 09 2011
9*n*(n-6)*a(n) + 2*(-17n^2+90n-133)*a(n-2) - 4*(n-4)(2n-9)*a(n-2) = 0. - R. J. Mathar, Nov 26 2011
G.f.: 64*x^6*(2*x+3*sqrt(1-4x))/( (1+sqrt(1-4x))^6 * (1-4x)^(3/2)). - R. J. Mathar, Nov 27 2011
a(n) ~ 4^n*sqrt(n)/(64*sqrt(Pi)). - Ilya Gutkovskiy, Apr 11 2017

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

N. J. A. Sloane, Jul 03 2015

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

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.

Diagonals give A002057, A002058, A002059, A002060.

Row sums give A065096 (with a different offset).

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

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 *)

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 */

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

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

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A002059 Number of partitions of an n-gon into (n-4) parts.

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

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