A253284 - OEIS

%I #26 Sep 08 2022 08:46:10

%S 1,2,2,6,18,12,24,144,240,120,120,1200,3600,4200,1680,720,10800,50400,

%T 100800,90720,30240,5040,105840,705600,2116800,3175200,2328480,665280,

%U 40320,1128960,10160640,42336000,93139200,111767040,69189120,17297280

%N Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.

%C G_n(x) = - Sum_{k=0..n} T(n,k)/(x-1)^(n+k+1) are generating functions, for n=0 of A000012, for n=1 of A002378, for n=2 of A083374 (with offset 0) and for n=3 for A253285. In general G_n(x) is the generating function of the sequence k -> ((n+k)!/k!)*C(n+k-1,k-1). These sequences are associated with the rows of the square array of unsigned Lah numbers (compare A253283 for the columns).

%F T(n,k) = (n+1)!*binomial(n+k,n)*binomial(n,k)/(k+1).

%F T(n,k) = (n+1)!*A088617(n,k).

%F T(n,0) = n! = A000142(n).

%F T(n,1) = A001804(n+1) for n>0.

%F T(n,n) = (2*n)!/n! = A001813(n).

%F Sum_{k=0..n} T(n,k) = (n+1)!*hypergeom([-n, n+1], [2], -1) = (n+1)!*A006318(n).

%e Triangle begins:

%e 1;

%e 2, 2;

%e 6, 18, 12;

%e 24, 144, 240, 120;

%e 120, 1200, 3600, 4200, 1680;

%e 720, 10800, 50400, 100800, 90720, 30240;

%e 5040, 105840, 705600, 2116800, 3175200, 2328480, 665280.

%p T := (n,k) -> ((k+1)*(n+1)!*(n+k)!)/((k+1)!^2*(n-k)!);

%p for n from 0 to 6 do seq(T(n,k), k=0..n) od;

%t f[n_] := Rest@ Flatten@ Reap@ Block[{i, k, t}, For[i = 0, i <= n, i++, For[k = 0, k <= i, k++, Sow[(i + 1)!*Binomial[i + k, i]*Binomial[i, k]/(k + 1)]]]]; f@ 7 (* _Michael De Vlieger_, Mar 23 2015 *)

%o (PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1((n+1)!*binomial(n+k,n)*binomial(n,k)/(k+1), ", ");); print(););} \\ _Michel Marcus_, Mar 23 2015

%o (Magma) /* As triangle: */ [[(k + 1)*Factorial(n + 1)*Factorial(n + k)/(Factorial(k + 1)^2*Factorial(n - k)): k in [0..n]]: n in [0..10]]; // _Bruno Berselli_, Mar 23 2015

%Y Cf. A000142, A001804, A001813, A002378, A006318, A083374, A088617, A253283, A253285.

%K nonn,tabl,easy

%O 0,2

%A _Peter Luschny_, Mar 23 2015