A268434 -id:A268434 - cp's OEIS frontend

A269938 Row sums of A268434.

1, 1, 3, 21, 269, 5501, 164871, 6826353, 373877161, 26202082729, 2288805048251, 243958190401341, 31176599353144853, 4706999020319020101, 829102910675624591839, 168548158717259440652601, 39174415082612704149839761, 10324151597447856368055425553, 3062691062053211798175516784691

Offset: 0

Peter Luschny, Mar 08 2016

nonn

Cf. A268434.

A269938 := proc(n) local T; T := proc(n, k) option remember;
if n=k then 1 elif k<0 or k>n then 0 else T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k) fi end: add(T(n, k), k=0..n) end: seq(A269938(n), n=0..18);

A269946 Triangle read by rows, Lah numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^3+k^3)*T(n-1, k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 18, 18, 1, 0, 504, 648, 72, 1, 0, 32760, 47160, 7200, 200, 1, 0, 4127760, 6305040, 1141560, 45000, 450, 1, 0, 895723920, 1416456720, 283704120, 13741560, 198450, 882, 1, 0, 308129028480, 498072032640, 106386981120, 5876519040, 106616160, 691488, 1568, 1

Offset: 0

Peter Luschny, Mar 22 2016

			Triangle starts:
[1]
[0, 1]
[0, 2,       1]
[0, 18,      18,      1]
[0, 504,     648,     72,      1]
[0, 32760,   47160,   7200,    200,   1]
[0, 4127760, 6305040, 1141560, 45000, 450, 1]

Peter Luschny, The P-transform.

Cf. A038207 (order 0), A111596 (order 1), A268434 (order 2).

Cf. A255433, A163102, A269947, A269948.

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + ((n-1)^3+k^3) * T(n-1, k) )) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^3 + k^3)*T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

T(n,k) = Sum_{j=k..n} A269947(n,j)*A269948(j,k).
T(n,1) = Product_{k=1..n} (k-1)^3+1 for n>=1 (cf. A255433).
T(n,n-1) = (n-1)^2*n^2/2 for n>=1 (cf. A163102).

cp's OEIS Frontend

A269938 Row sums of A268434.

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A269946 Triangle read by rows, Lah numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^3+k^3)*T(n-1, k), for n>=0 and 0<=k<=n.

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