A generating function for the diagonal \(T_{2n,n}\) in triangles. (English) Zbl 1327.05018
Summary: We present techniques for obtaining a generating function for the diagonal \(T_{2n,n}\) of the triangle formed from the coefficients of a generating function \(G(x)\) raised to the power \(k\). We obtain some relations between central coefficients and coefficients of the diagonal \(T_{2n,n}\), and we also give some examples.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 11B75 | Other combinatorial number theory |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
Software:
OEISOnline Encyclopedia of Integer Sequences:
Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.a(n) = binomial(3*n,n)/(2*n+1) (enumerates ternary trees and also noncrossing trees).
a(n) = [x^(2*n)] ((1 + x)/(1 - x))^n.
a(n) = binomial(5*n-1,n).
Triangle read by rows, based on expansion of (x^2/(exp(x)-1))^m = x^m+sum(n>m T(n,m)*m!/((n-m)!*n!)*x^n).
Triangle T(n,m) = coefficient of x^n in expansion of (x^2*cotan(x))^m = sum(n>=m, T(n,m) x^n * m!^2/n!^2).
a(n) = C(5*n-1,2*n)/3, n > 0, a(0) = 1.
a(n) = Sum_{i=0..n} (2^(i)*(-1)^(i+n)*C(n,i)*C(2*n+i-1,n-1)).
a(n) = binomial(4*n-1, 2*n).
a(n) = binomial(6*n,2*n)/3, n>0, a(0)=1.
a(n) = binomial(4*n-1,n).
