A073474 Triangle T(n,k) read by rows, where o.g.f. for T(n,k) is n!*Sum_{k=0..n} (1+x)^(n-k)/k!.

1, 2, 1, 5, 6, 2, 16, 33, 24, 6, 65, 196, 228, 120, 24, 326, 1305, 2120, 1740, 720, 120, 1957, 9786, 20550, 23160, 14760, 5040, 720, 13700, 82201, 212352, 305970, 265440, 138600, 40320, 5040, 109601, 767208, 2356424, 4146576, 4571280, 3232320, 1431360, 362880, 40320

Offset

0,2

Comments

Row sums give A010844.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

E.g.f.: exp(x)/(1-x-x*y). - Vladeta Jovovic, Oct 17 2003

Example

Triangle begins:
    1;
    2,    1;
    5,    6,    2;
   16,   33,   24,    6;
   65,  196,  228,  120,  24;
  326, 1305, 2120, 1740, 720, 120;
  ...

Maple

G:=simplify(series(exp(x)/(1-x-x*y), x=0, 13)): P[0]:=1: for n from 1 to 11 do P[n]:=sort(n!*coeff(G, x^n)) od: seq(seq(coeff(y*P[n], y^k), k=1..n+1), n=0..9);
# second Maple program:
b:= proc(n, k) option remember; `if`(k>n, 0, `if`(k=0, 1,
      n*(b(n-1, k-1)+b(n-1, k))))
    end:
T:= (n, k)-> b(n+1, k+1)/(n+1):
seq(seq(T(n, k), k=0..n), n=0..9);  # Alois P. Heinz, Sep 12 2019

Mathematica

b[n_, k_] := b[n, k] = If[k>n, 0, If[k==0, 1, n (b[n-1, k-1]+b[n-1, k])]];
T[n_, k_] := b[n+1, k+1]/(n+1);
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 09 2019, after Alois P. Heinz *)

Crossrefs

Cf. A000142, A000522, A073107.

Sequence in context: A234950 A275228 A118984 * A067311 A341487 A162750

Adjacent sequences: A073471 A073472 A073473 * A073475 A073476 A073477

Keyword

easy,nonn,tabl

Author

Vladeta Jovovic, Aug 26 2002

Extensions

Edited by Emeric Deutsch, Jun 10 2004

Status

approved