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Cayley's triangle of V numbers; triangle V(n,k), n >= 4, n <= k <= 2*n-4, read by rows.
1

%I #28 Apr 15 2023 02:08:42

%S 1,2,4,3,14,14,4,32,72,48,5,60,225,330,165,6,100,550,1320,1430,572,7,

%T 154,1155,4004,7007,6006,2002,8,224,2184,10192,25480,34944,24752,7072,

%U 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

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

%H A. Cayley, <a href="http://dx.doi.org/10.1112/plms/s1-22.1.237">On the partitions of a polygon</a>, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.

%F 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

%F T(n,m) = 2*C(m,n)*C(n-2,m-n+2)/(n-2), n>=4. - _Vladimir Kruchinin_, Jan 27 2022

%e Triangle begins:

%e 1;

%e 2, 4;

%e 3, 14, 14;

%e 4, 32, 72, 48;

%e 5, 60, 225, 330, 165;

%e 6, 100, 550, 1320, 1430, 572;

%e ...

%p V := proc(n,x)

%p local X,g,i ;

%p X := x^2/(1-x) ;

%p g := X^n ;

%p for i from 1 to n-2 do

%p g := diff(g,x) ;

%p end do;

%p x^2*g*2*(n-1)/n! ;

%p end proc;

%p A259476 := proc(n,k)

%p V(k-n+2,x) ;

%p coeftayl(%,x=0,n+2) ;

%p end proc:

%p for n from 4 to 14 do

%p for k from n to 2*n-4 do

%p printf("%d,",A259476(n,k)) ;

%p end do:

%p printf("\n") ;

%p end do: # _R. J. Mathar_, Jul 09 2015

%t T[n_, m_] := 2 Binomial[m, n] Binomial[n-2, m-n+2]/(n-2);

%t Table[T[n, m], {n, 4, 14}, {m, n, 2n-4}] // Flatten (* _Jean-François Alcover_, Apr 15 2023, after _Vladimir Kruchinin_ *)

%o (Maxima)

%o 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 */

%Y Diagonals give A002057, A002058, A002059, A002060.

%Y Row sums give A065096 (with a different offset).

%K nonn,tabl

%O 4,2

%A _N. J. A. Sloane_, Jul 03 2015