

A320657


a(n) is the number of nonunimodal sequences with n nonzero terms that arise as a convolution of sequences of binomial coefficients preceded by a finite number of ones.


0



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, 185, 210, 245, 275, 316, 352, 400, 442, 497, 546, 609, 665, 736, 800, 880, 952, 1041, 1122, 1221, 1311, 1420, 1520, 1640, 1750, 1881, 2002, 2145, 2277, 2432, 2576, 2744, 2900, 3081, 3250, 3445, 3627, 3836, 4032
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OFFSET

1,12


COMMENTS

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 nonunimodal sequences (r_i)_{i>=1} where r_i = Sum_{j=1..i} s_j*t_{ij} for some(s_i) and (t_i) such that x + y + p + q + 1 = n.
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, vertexinduced subgraphs is not rank unimodal.


LINKS

Table of n, a(n) for n=1..64.
T. M. Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018). See table 1 on page 17.
M. Jacobson, A. E. Kézdy, and S. Seif, The poset on connected induced subgraphs of a graph need not be Sperner, Order, 12 (1995) 315318.


FORMULA

a(n+10) = 2*(Sum_{i=1..n/2} floor(i*(i+4)/4))  floor(n^2/16) for n even.
a(n+10) = 2*(Sum_{i=1..(n1)/2} floor(i(i+4)/4))  floor((n1)^2/16) + floor((n+1)*(n+9)/16) for n odd.


MATHEMATICA

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, (n1)/2}])  Floor[(n1)^2/16] + Floor[(n+1)(n+9)/16]], {n, 0, 40}]


CROSSREFS

Cf. A005993, A024206. Equals A005581 for n even.
Sequence in context: A161664 A080547 A080555 * A024924 A023668 A023564
Adjacent sequences: A320654 A320655 A320656 * A320658 A320659 A320660


KEYWORD

nonn


AUTHOR

Tricia Muldoon Brown, Oct 17 2018


STATUS

approved



