A206464 Number of length-n Catalan-RGS (restricted growth strings) such that the RGS is a valid mixed-radix number in falling factorial basis.

1, 1, 2, 4, 10, 26, 74, 218, 672, 2126, 6908, 22876, 77100, 263514, 911992, 3189762, 11261448, 40083806, 143713968, 518594034, 1882217168, 6867064856, 25172021144, 92666294090, 342467464612, 1270183943200, 4726473541216, 17640820790092, 66025467919972

Offset

0,3

Comments

Catalan-RGS are strings with first digit d(0)=zero, and d(k+1) <= d(k)+1, falling factorial mixed-radix numbers have last digit <= 1, second last <= 2, etc.

The digits of the RGS are <= floor(n/2).

The first few terms are the same as for A089429.

Column k=0 of A264869. - Peter Bala, Nov 27 2015

a(n) = A291680(n+1,n+1). - Alois P. Heinz, Aug 29 2017

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

Conjecture: a(n) = Sum_{k = 0..floor(n/4)} (-1)^k * C(floor(n/2) + 1 - k, k + 1) * a(n - 1 - k), a(0) = 1. - Gionata Neri, Jun 17 2018

Example

The a(5)=26 strings for n=5 are (dots for zeros):
   1:  [ . . . . . ]
   2:  [ . . . . 1 ]
   3:  [ . . . 1 . ]
   4:  [ . . . 1 1 ]
   5:  [ . . 1 . . ]
   6:  [ . . 1 . 1 ]
   7:  [ . . 1 1 . ]
   8:  [ . . 1 1 1 ]
   9:  [ . . 1 2 . ]
  10:  [ . . 1 2 1 ]
  11:  [ . 1 . . . ]
  12:  [ . 1 . . 1 ]
  13:  [ . 1 . 1 . ]
  14:  [ . 1 . 1 1 ]
  15:  [ . 1 1 . . ]
  16:  [ . 1 1 . 1 ]
  17:  [ . 1 1 1 . ]
  18:  [ . 1 1 1 1 ]
  19:  [ . 1 1 2 . ]
  20:  [ . 1 1 2 1 ]
  21:  [ . 1 2 . . ]
  22:  [ . 1 2 . 1 ]
  23:  [ . 1 2 1 . ]
  24:  [ . 1 2 1 1 ]
  25:  [ . 1 2 2 . ]
  26:  [ . 1 2 2 1 ]

Maple

b:= proc(i, l) option remember;
      `if`(i<=0, 1, add(b(i-1, j), j=0..min(l+1, i)))
    end:
a:= n-> b(n-1, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 08 2012

Mathematica

b[i_, l_] := b[i, l] = If[i <= 0, 1, Sum[b[i-1, j], {j, 0, Min[l+1, i]}]];
a[n_] := b[n-1, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

Crossrefs

Cf. A080935, A080936, A264869, A291680.

Sequence in context: A245898 A230662 A351642 * A089429 A049143 A089404

Adjacent sequences: A206461 A206462 A206463 * A206465 A206466 A206467

Keyword

nonn

Author

Joerg Arndt, Feb 08 2012

Status

approved