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A046740 Triangle of number of permutations of [n] with 0 successions, by number of rises. 3
1, 1, 1, 2, 1, 8, 2, 1, 22, 28, 2, 1, 52, 182, 72, 2, 1, 114, 864, 974, 164, 2, 1, 240, 3474, 8444, 4174, 352, 2, 1, 494, 12660, 57194, 61464, 15782, 732, 2, 1, 1004, 43358, 332528, 660842, 373940, 55286, 1496, 2, 1, 2026, 142552, 1747558, 5814124 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The recurrence given by Roselle is wrong.

LINKS

Table of n, a(n) for n=1..51.

D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16.

D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]

FORMULA

a(n, 1) = 1; for r > 1, a(n, r) = r*a(n-1, r) + (n-r)*a(n-1, r-1) + (n-2)*a(n-2, r-1).

a(n, 2) = 2^n - 2*n = 2*A000295 = A005803, n >= 3.

EXAMPLE

Triangle begins:

  1;

  1;

  1,  2;

  1,  8,  2;

  1, 22, 28,  2;

  ...

MATHEMATICA

a[_, 1] = 1; a[n_, 2] := 2^n - 2*n; a[n_, r_] /; 1 <= r <= n-1 := a[n, r] = r*a[n-1, r] + (n-r)*a[n-1, r-1] + (n-2)*a[n-2, r-1]; a[_, _] = 0;

row[1] = {{1}}; row[n_] := Table[a[n, r], {r, 1, n-1}];

Table[row[n], {n, 1, 11}] // Flatten (* Jean-Fran├žois Alcover, Sep 07 2017 *)

CROSSREFS

Cf. A046739, A000295. Row sums give A000255. Diagonals give A005803, A065340.

Row sums give A000255.

Sequence in context: A208921 A208660 A114706 * A317932 A253583 A130562

Adjacent sequences:  A046737 A046738 A046739 * A046741 A046742 A046743

KEYWORD

nonn,easy,nice,tabf

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Jan 03 2003

STATUS

approved

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Last modified May 12 10:13 EDT 2021. Contains 343821 sequences. (Running on oeis4.)