A143569 -id:A143569 - cp's OEIS frontend

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

Alois P. Heinz, Aug 24 2008

			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;
  ...

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

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

T(2n,n) gives A319924.

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);

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 *)

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

A367757 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2 + x^3 + x^4) * A(x^5/120)).

Original entry on oeis.org

1, 1, 3, 13, 73, 501, 3337, 27637, 254409, 2557369, 27603631, 313768731, 3905502745, 51573777841, 718307494269, 10507900625251, 161239887204721, 2608009648536417, 43989477103304155, 772109936171046001, 14085074476090366761, 266890182641557777093

Offset: 0

Seiichi Manyama, Nov 29 2023

nonn

Cf. A367754, A367755, A367756.

Cf. A143569.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, i-1, (j+1)*v[j\5+1]*v[i-j]/(120^(j\5)*(j\5)!*(i-1-j)!))); v;

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..n-1} (k+1) * a(floor(k/5)) * a(n-1-k) / (120^floor(k/5) * floor(k/5)! * (n-1-k)!).

cp's OEIS Frontend

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!)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula