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

Author

N. J. A. Sloane, Jul 03 2015

Keywords

• nonn
• tabl

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