A143565 Triangle T(n,k), n>=1, 1<=k<=n, where the e.g.f. for column k satisfies: A_k(x) = exp(x*A_k(x^k/k!)).

1, 3, 1, 16, 4, 1, 125, 13, 5, 1, 1296, 46, 21, 6, 1, 16807, 241, 61, 31, 7, 1, 262144, 1471, 211, 106, 43, 8, 1, 4782969, 9409, 1401, 281, 169, 57, 9, 1, 100000000, 67348, 8065, 946, 505, 253, 73, 10, 1, 2357947691, 564841, 37241, 7561, 1261, 841, 361, 91, 11, 1

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..100, flattened

Formula

E.g.f. for column k satisfies: A_k(x) = exp(x*A_k(x^k/k!)).

T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=0..floor((n-1)/k)} (k*j+1) * T(j,k) * T(n-1-k*j,k) / (k!^j * j! * (n-1-k*j)!). - Seiichi Manyama, Nov 28 2023

Example

Triangle begins:
       1;
       3,    1;
      16,    4,    1;
     125,   13,    5,    1;
    1296,   46,   21,    6,    1;
   16807,  241,   61,   31,    7,    1;
  262144, 1471,  211,  106,   43,    8,    1;
  ...

Maple

A:= proc(n, k::posint) option remember; if n<=0 then 1 else unapply(
      convert(series(exp(x*A(n-k, k)(x^k/k!)), x, n+1), polynom), x) fi
    end:
T:= (n, k)-> coeff(A(n, k)(x), x, n)*n!:
seq(seq(T(n, k), k=1..n), n=1..12);

Mathematica

a[n_, k_] := a[n, k] = If[n <= 0, 1&, Function[x, Series[E^(x*a[n - k, k][x^k/k!]), {x, 0, n+1}] // Normal // Evaluate]]; t[n_, k_] := Coefficient[a[n, k][x], x, n]*n!; Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, 12}]] (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

Crossrefs

Columns 1-9: A000272, A143566, A143567, A143568, A143569, A143570, A143571, A143572, A143573.

T(2n,n) gives A319924.

Sequence in context: A362847 A168319 A326374 * A143018 A102012 A128249

Adjacent sequences: A143562 A143563 A143564 * A143566 A143567 A143568

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 24 2008

Status

approved