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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.

%I A259476 #28 Apr 15 2023 02:08:42
%S A259476 1,2,4,3,14,14,4,32,72,48,5,60,225,330,165,6,100,550,1320,1430,572,7,
%T A259476 154,1155,4004,7007,6006,2002,8,224,2184,10192,25480,34944,24752,7072,
%U A259476 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 A259476 Cayley's triangle of V numbers; triangle V(n,k), n >= 4, n <= k <= 2*n-4, read by rows.
%H A259476 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 A259476 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 A259476 T(n,m) = 2*C(m,n)*C(n-2,m-n+2)/(n-2), n>=4. - _Vladimir Kruchinin_, Jan 27 2022
%e A259476 Triangle begins:
%e A259476   1;
%e A259476      2, 4;
%e A259476         3, 14, 14;
%e A259476             4, 32, 72,  48;
%e A259476                 5, 60, 225, 330,  165;
%e A259476                     6, 100, 550, 1320, 1430, 572;
%e A259476   ...
%p A259476 V := proc(n,x)
%p A259476     local X,g,i ;
%p A259476     X := x^2/(1-x) ;
%p A259476     g := X^n ;
%p A259476     for i from 1 to n-2 do
%p A259476         g := diff(g,x) ;
%p A259476     end do;
%p A259476     x^2*g*2*(n-1)/n! ;
%p A259476 end proc;
%p A259476 A259476 := proc(n,k)
%p A259476     V(k-n+2,x) ;
%p A259476     coeftayl(%,x=0,n+2) ;
%p A259476 end proc:
%p A259476 for n from 4 to 14 do
%p A259476     for k from n to 2*n-4 do
%p A259476         printf("%d,",A259476(n,k)) ;
%p A259476     end do:
%p A259476     printf("\n") ;
%p A259476 end do: # _R. J. Mathar_, Jul 09 2015
%t A259476 T[n_, m_] := 2 Binomial[m, n] Binomial[n-2, m-n+2]/(n-2);
%t A259476 Table[T[n, m], {n, 4, 14}, {m, n, 2n-4}] // Flatten (* _Jean-François Alcover_, Apr 15 2023, after _Vladimir Kruchinin_ *)
%o A259476 (Maxima)
%o A259476 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 A259476 Diagonals give A002057, A002058, A002059, A002060.
%Y A259476 Row sums give A065096 (with a different offset).
%K A259476 nonn,tabl
%O A259476 4,2
%A A259476 _N. J. A. Sloane_, Jul 03 2015