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

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

4,2

Links

Table of n, a(n) for n=4..69.

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.

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

Example

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

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 *)

Prog

(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 */

Crossrefs

Diagonals give A002057, A002058, A002059, A002060.

Row sums give A065096 (with a different offset).

Sequence in context: A079308 A189825 A271878 * A271363 A115399 A109429

Adjacent sequences: A259473 A259474 A259475 * A259477 A259478 A259479

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jul 03 2015

Status

approved