A028309 -id:A028309 - cp's OEIS frontend

A379202 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).

1, 4, 20, 122, 850, 6432, 51324, 424694, 3608592, 31291658, 275774228, 2462835772, 22239367632, 202713590686, 1862689951724, 17235880764264, 160466865121154, 1502055108051124, 14127846520455180, 133455751612975948, 1265563747442829216, 12043611154775588194, 114978748131733714360

Offset: 1

Paul D. Hanna, Dec 20 2024

nonn

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.
Conjecture: a(n) is even for n > 1.
It appears that a(n) == 2 (mod 4) at n = A028309(k) for k >= 4.

			G.f.: A(x) = x + 4*x^2 + 20*x^3 + 122*x^4 + 850*x^5 + 6432*x^6 + 51324*x^7 + 424694*x^8 + 3608592*x^9 + 31291658*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.090270773138940793847220645261976952310511883470512...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*6^(n-1))^n / 6^(n^2-1).
A(t) = 1/7 at t = 0.084362907984862824662513569761745773472320783010611...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*7^(n-1))^n / 7^(n^2-1).
A(t) = 1/8 at t = 0.078703999402417120618295617221021413542415048822164...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*8^(n-1))^n / 8^(n^2-1).
A(1/11) = 0.16976727159020613475135380983780463368461713164010...
A(1/12) = 0.13933682309394427848416123650354034389806333559384...
A(1/15) = 0.09515898887066227963795425335824195002284059150209...
A(1/20) = 0.06369786461564277053938913595571090186089127528505...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A379200, A379199, A166952, A378264, A379203, A379204, A379205.

{a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 2)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 2)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 2*A(x)^(n+1))^n.
(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 2*A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 2)^(n+1) ).

A363364 Least nonnegative integer k such that all non-bipartite graphs with n nodes and at least k edges are weakly pancyclic.

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 11, 14, 17, 20

Offset: 1

Pontus von Brömssen, May 29 2023

A graph is weakly pancyclic if it contains cycles of all lengths between its girth and its circumference. Acyclic graphs are considered to be weakly pancyclic.
All graphs on at most 5 nodes are weakly pancyclic, so a(n) = 0 when n <= 5.
Brandt (1997) conjectures that a(n) = floor((n-1)*(n-3)/4) + 5 for n >= 6. The conjecture is false for n = 8, since there exists a (unique) non-bipartite, not weakly pancyclic graph (shown below) with 8 nodes and 13 edges, showing that a(8) >= 14. This graph contains cycles of lengths 3, 4, 5, 6, and 8, but none of length 7.
         O
        /|\
       / O \
      /  |  \
     /   O   \
    /  /   \  \
   / /       \ \
  //           \\
  O ----------- O
  \\           //
   \ \       / /
    \  \   /  /
     \   O   /
      \  |  /
       \ O /
        \|/
         O

Béla Bollobás and Andrew Thomason, Weakly pancyclic graphs, Journal of Combinatorial Theory Series B 77 (1999), 121-137.
Stephan Brandt, A sufficient condition for all short cycles, Discrete Applied Mathematics 79 (1997), 63-66.

Cf. A028309, A290743, A363362, A363363.

a(n) >= floor((n-1)*(n-3)/4) + 5 = A028309(n-1) + 2 for n >= 6 (Brandt, 1997).
a(n) <= floor((n-1)^2/4) + 2 = A290743(n-1) (Brandt, 1997).
a(n) <= floor(n^2/4) - n + 59 (Bollobás and Thomason, 1999).

cp's OEIS Frontend

A379202 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).

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