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 A263855 Number of connected graphs on n nodes up to isomorphism with a factor of (1+x) in their independence polynomial. 0
 1, 6, 38, 277, 3056, 59768, 2376028, 195245762, 31700259751 (list; graph; refs; listen; history; text; internal format)
 OFFSET 4,2 LINKS F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version b3ab217. EXAMPLE For n = 4, the a(4)=1 solution is the path of length 3. PROG (Sage) from sage.graphs.independent_sets import IndependentSets from math import factorial from time import time #Function to calculate a binomial coefficient (n choose r) def choose(n, r):     return factorial(n) / (factorial(r) * factorial(n - r)) #Function that checks if a polynomial has a certain root def root_in_poly(poly, root):     root_list = poly.roots()     for tuple in root_list:         for elt in tuple:             if root == elt:                 return True     return False #Builds an independence polynomial for a graph def build_ip(graph):     number_of = [0] * graph.order()     for set in IndependentSets(graph):         number_of[len_set] += 1;     poly = 0     for index in range(0, len(number_of)):         poly += (number_of[index]) * (x ** index)     return poly ip_list = [] R. = QQ[] root = -1 for v in range(4, 10):     count = 0     for g in graphs(v):         if g.is_connected():             ip = build_ip(g)             if root_in_poly(ip, root):                 ip_list.append(ip)                 count += 1     print(v, ": ", count) CROSSREFS Sequence in context: A026940 A082427 A192941 * A221283 A064309 A075197 Adjacent sequences:  A263852 A263853 A263854 * A263856 A263857 A263858 KEYWORD nonn,more AUTHOR Ethan J. Brockmann, Nov 03 2015 EXTENSIONS a(10)-a(12) added using tinygraph by Falk Hüffner, Jan 20 2016 STATUS approved

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Last modified October 21 06:36 EDT 2019. Contains 328292 sequences. (Running on oeis4.)