cp's OEIS Frontend

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006

Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012

Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012

A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x). See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012

The matrix inverse starts

1;

-1, 1;

-1, -1, 1;

1, -2, -1, 1;

3, 1, -3, -1, 1;

1, 6, 1, -4, -1, 1;

-7, 4, 10, 1, -5, -1, 1;

-13, -13, 8, 15, 1, -6, -1, 1;

3, -31, -23, 13, 21, 1, -7, -1, 1; - R. J. Mathar, Mar 15 2013

Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025

This constant is Heuberger and Wagner's lambda. They consider the number of maximum matchings a tree of n vertices may have, and show that the largest number of maximum matchings (A333347) grows as O(lambda^(n/7)) (see A333346 for the 7th root). Lambda is the larger eigenvalue of matrix M = [8,3/5,3] which is raised to a power when counting matchings in a chain of "C" parts in the trees (their lemma 6.2).

Apart from the first digit the same as A176522. - R. J. Mathar, Apr 03 2020

Heuberger and Wagner consider how many different maximum matchings a tree of n vertices may have. They determine the unique tree (free tree) of n vertices with the largest number of maximum matchings, or at n=6 and n=34 the two trees with equal largest number. a(n) is the largest number of maximum matchings. They show that a(n) grows as O(1.391...^n), where the power is ((11 + sqrt(85))/2)^(1/7) = A333346.

They note an algebraic interpretation too, that a(n) is the largest possible absolute value of the product of the nonzero eigenvalues of the adjacency matrix of a tree of n vertices. This is simply that, in the usual way, a term +- m*x^j in the characteristic polynomial of that matrix means there are m matchings which have j vertices unmatched. The smallest j with a nonzero m is the maximum matchings, and that m is also the product of the nonzero roots.

In Heuberger and Wagner's Sage code, optimal_m(n) is a(n) for the general case tree forms. Their general case symbolic calculations are in terms of lambda = (11 + sqrt(85))/2 = A333345 and its quadratic conjugate lambdabar = (11 - sqrt(85))/2 (called alpha and alphabar in the code). The resulting coefficients give constants c_0 through c_6 in their paper for a(n) -> c_{n mod 7} * lambda^(n/7) (theorem 1.2).

The combinations of powers of lambda and lambdabar occurring are linear recurrences. Recurrence coefficients can be found from a symbolic calculation, or from explicit values and an upper bound on recurrence orders from the patterns of branch lengths and powers. Each case n mod 7 is a recurrence of order up to 44. The simplest is G_k = A190872(k) for n == 1 (mod 7) in the formulas below. Other cases are G variants, and possible additional terms growing slower than G.

The full recurrence for all n is order 574 applying at n=31 onwards (after the last initial exception at n=30). See the links for recurrence coefficients and generating function.

First differences of A190872.

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*b(n - 2), with n>1 and b(0)=0, b(1)=1, is x/(1 - k*x + m*x^2). This recurrence gives the closed form b(n) = (2^(-n)*((sqrt(k^2 - 4*m) + k)^n - (k - sqrt(k^2 - 4*m))^n))/sqrt(k^2 - 4*m).