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

%I A206464 #43 Mar 09 2024 14:23:40
%S A206464 1,1,2,4,10,26,74,218,672,2126,6908,22876,77100,263514,911992,3189762,
%T A206464 11261448,40083806,143713968,518594034,1882217168,6867064856,
%U A206464 25172021144,92666294090,342467464612,1270183943200,4726473541216,17640820790092,66025467919972
%N A206464 Number of length-n Catalan-RGS (restricted growth strings) such that the RGS is a valid mixed-radix number in falling factorial basis.
%C A206464 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.
%C A206464 The digits of the RGS are <= floor(n/2).
%C A206464 The first few terms are the same as for A089429.
%C A206464 Column k=0 of A264869. - _Peter Bala_, Nov 27 2015
%C A206464 a(n) = A291680(n+1,n+1). - _Alois P. Heinz_, Aug 29 2017
%H A206464 Alois P. Heinz, <a href="/A206464/b206464.txt">Table of n, a(n) for n = 0..1000</a>
%F A206464 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
%e A206464 The a(5)=26 strings for n=5 are (dots for zeros):
%e A206464    1:  [ . . . . . ]
%e A206464    2:  [ . . . . 1 ]
%e A206464    3:  [ . . . 1 . ]
%e A206464    4:  [ . . . 1 1 ]
%e A206464    5:  [ . . 1 . . ]
%e A206464    6:  [ . . 1 . 1 ]
%e A206464    7:  [ . . 1 1 . ]
%e A206464    8:  [ . . 1 1 1 ]
%e A206464    9:  [ . . 1 2 . ]
%e A206464   10:  [ . . 1 2 1 ]
%e A206464   11:  [ . 1 . . . ]
%e A206464   12:  [ . 1 . . 1 ]
%e A206464   13:  [ . 1 . 1 . ]
%e A206464   14:  [ . 1 . 1 1 ]
%e A206464   15:  [ . 1 1 . . ]
%e A206464   16:  [ . 1 1 . 1 ]
%e A206464   17:  [ . 1 1 1 . ]
%e A206464   18:  [ . 1 1 1 1 ]
%e A206464   19:  [ . 1 1 2 . ]
%e A206464   20:  [ . 1 1 2 1 ]
%e A206464   21:  [ . 1 2 . . ]
%e A206464   22:  [ . 1 2 . 1 ]
%e A206464   23:  [ . 1 2 1 . ]
%e A206464   24:  [ . 1 2 1 1 ]
%e A206464   25:  [ . 1 2 2 . ]
%e A206464   26:  [ . 1 2 2 1 ]
%p A206464 b:= proc(i, l) option remember;
%p A206464       `if`(i<=0, 1, add(b(i-1, j), j=0..min(l+1, i)))
%p A206464     end:
%p A206464 a:= n-> b(n-1, 0):
%p A206464 seq(a(n), n=0..40);  # _Alois P. Heinz_, Feb 08 2012
%t A206464 b[i_, l_] := b[i, l] = If[i <= 0, 1, Sum[b[i-1, j], {j, 0, Min[l+1, i]}]];
%t A206464 a[n_] := b[n-1, 0];
%t A206464 a /@ Range[0, 40] (* _Jean-François Alcover_, Nov 07 2020, after _Alois P. Heinz_ *)
%Y A206464 Cf. A080935, A080936, A264869, A291680.
%K A206464 nonn
%O A206464 0,3
%A A206464 _Joerg Arndt_, Feb 08 2012