OFFSET
4,2
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.
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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jul 03 2015
STATUS
approved