A320657 - OEIS

%I #25 Dec 14 2018 20:54:33

%S 0,0,0,0,0,0,0,0,0,0,1,2,5,7,12,16,24,30,41,50,65,77,96,112,136,156,

%T 185,210,245,275,316,352,400,442,497,546,609,665,736,800,880,952,1041,

%U 1122,1221,1311,1420,1520,1640,1750,1881,2002,2145,2277,2432,2576,2744,2900,3081,3250,3445,3627,3836,4032

%N a(n) is the number of non-unimodal sequences with n nonzero terms that arise as a convolution of sequences of binomial coefficients preceded by a finite number of ones.

%C For integers x,y,p,q >= 0, set (s_i)_{i>=1} to be the sequence of p ones followed by the binomial coefficients C(x,j) for 0 <= j <= x followed by an infinite string of zeros, and set (t_i)_{i>=1} to be the sequence of q ones followed by the binomial coefficients C(y,j) for 0 <= j <= y followed by an infinite string of zeros. Then a(n) is the number of non-unimodal sequences (r_i)_{i>=1} where r_i = Sum_{j=1..i} s_j*t_{i-j} for some(s_i) and (t_i) such that x + y + p + q + 1 = n.

%C Let T be a rooted tree created by identifying the root vertices of two broom graphs. a(n) is the number of trees T on n vertices whose poset of connected, vertex-induced subgraphs is not rank unimodal.

%H T. M. Brown, <a href="https://arxiv.org/abs/1810.08235">On the unimodality of convolutions of sequences of binomial coefficients</a>, arXiv:1810.08235 [math.CO] (2018). See table 1 on page 17.

%H M. Jacobson, A. E. Kézdy, and S. Seif, <a href="https://doi.org/10.1007/BF01111746">The poset on connected induced subgraphs of a graph need not be Sperner</a>, Order, 12 (1995) 315-318.

%F a(n+10) = 2*(Sum_{i=1..n/2} floor(i*(i+4)/4)) - floor(n^2/16) for n even.

%F a(n+10) = 2*(Sum_{i=1..(n-1)/2} floor(i(i+4)/4)) - floor((n-1)^2/16) + floor((n+1)*(n+9)/16) for n odd.

%t Table[If[EvenQ[n], 2*(Sum[Floor[i(i+4)/4], {i,0,(n/2)}]) - Floor[n^2/16], 2*(Sum[Floor[i(i+4)/4], {i,0,(n-1)/2}]) - Floor[(n-1)^2/16] + Floor[(n+1)(n+9)/16]], {n,0,40}]

%Y Cf. A005993, A024206. Equals A005581 for n even.

%K nonn

%O 1,12

%A _Tricia Muldoon Brown_, Oct 17 2018