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A294281 Number of ascent sequences of length n with alternating ascents and descents (unaffected by level steps).

%I A294281 #12 Oct 27 2017 07:43:50
%S A294281 1,1,2,4,9,22,59,172,547,1886,7047,28360,122675,567210,2796999,
%T A294281 14641044,81191947,475148678,2929442263,18965690560,128754649699,
%U A294281 914056305794,6777666961735,52367331911180,421188392986843,3519168714308702,30519733808467031
%N A294281 Number of ascent sequences of length n with alternating ascents and descents (unaffected by level steps).
%H A294281 Alois P. Heinz, <a href="/A294281/b294281.txt">Table of n, a(n) for n = 0..550</a>
%F A294281 a(n) = Sum_{j=0..n} binomial(n-1,j) * A099960(n-j).
%e A294281 a(3) = 4: 000, 001, 010, 011.
%e A294281 a(4) = 9: 0000, 0001, 0010, 0011, 0100, 0101, 0102, 0110, 0111.
%e A294281 a(5) = 22: 00000, 00001, 00010, 00011, 00100, 00101, 00102, 00110, 00111, 01000, 01001, 01002, 01010, 01011, 01020, 01021, 01022, 01100, 01101, 01102, 01110, 01111.
%p A294281 b:= proc(n, i, t, u) option remember; `if`(n<1, 1, add(
%p A294281        b(n-1, j, t+`if`(j>i, 1, 0), `if`(i=j, u, 1-u)),
%p A294281        j=`if`(u=0, i..t+1, 0..i)))
%p A294281     end:
%p A294281 a:= n-> b(n-1, 0$3):
%p A294281 seq(a(n), n=0..30);
%Y A294281 Cf. A022493, A099960.
%K A294281 nonn
%O A294281 0,3
%A A294281 _Alois P. Heinz_, Oct 26 2017