A111596 -id:A111596 - cp's OEIS frontend

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.

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).

A344051 a(n) = Sum_{k=0..n} binomial(n, k)*|Lah(n, k)|. Binomial convolution of the unsigned Lah numbers A271703.

Original entry on oeis.org

1, 1, 5, 37, 361, 4301, 60001, 954325, 16984577, 333572041, 7151967181, 165971975621, 4139744524345, 110333560295557, 3126749660583641, 93819198847833061, 2969676820062708481, 98843743790129998865, 3449675368718647501717, 125921086600579132143781, 4796519722094585691925961

Offset: 0

Peter Luschny, May 10 2021

nonn

Cf. A271703, A111596, A344050.

aList := proc(len) local lah;
lah := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1)*n!/k!):
seq(add(binomial(n, k)*lah(n, k), k = 0..n), n = 0..len-1) end:
lprint(aList(22));

a[n_] := n n! HypergeometricPFQ[{1 - n, 1 - n}, {2, 2}, 1]; a[0] := 1;
Table[a[n], {n, 0, 20}]

a(n) = n * n! * hypergeom([1 - n, 1 - n], [2, 2], 1) for n >= 1.
D-finite with recurrence +16*n*a(n) +6*(-8*n^2+5*n-1)*a(n-1) +(48*n^3-266*n^2+407*n-167)*a(n-2) +(-16*n^4+106*n^3-219*n^2+108*n+93)*a(n-3) +(n-4)*(2*n^3-13*n^2+16*n+25)*a(n-4) -(n-5)*(n-4)^3*a(n-5)=0. - R. J. Mathar, Jul 27 2022
a(n) ~ n^(n - 1/2) / (sqrt(6*Pi) * exp(n - 3*n^(2/3) + n^(1/3) - 1/3)) * (1 + 31/(54*n^(1/3))). - Vaclav Kotesovec, Apr 27 2024

cp's OEIS Frontend

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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