On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles. (English) Zbl 1108.11024
In this very well written paper, the author looks at the function
\[
B(x,y,t)={{x-y}\over {xe^{-{{(x-y)}\over {2}}t}-ye^{{{(x-y)}\over {2}}t}}}
\]
for \(x\neq y\) and \(1/(1-xt)\) for \(x=y\). He shows that if \(c\) is any complex number then
\[
B^c(x,y,t)=\sum_{n=0}^{\infty} 2^{-n} B_n(x,y,c) {{t^n}\over {n!}},
\]
converges absolutely for \(| t| <| (\ln x-\ln y)/(x-y)| \), where
\[
B_n(x,y,c)=\sum_{k=0}^n B_{n,k}(c)x^{n-k}y^k,
\]
and the coefficients \(B_{n,k}(c)\) satisfy the recurrence
\[
B_{n+1,k+1}(c)=(2(k+1)+c)B_{n,k+1}(c)+(2(n-k)+c)B_{n,k}(c),
\]
with \(B_{0,0}(c)=1\) and \(B_{n,k}(c)=0\) if \(k>n\). The author also studies various properties of the numbers \(B_{n,k}(c)\). By specializing to certain particular values of the parameters, the author finds that the numbers \(B_{n,k}(c)\) are linked to Eulerian numbers, MacMahon numbers and Stirling numbers and he derives various new identities involving these numbers.
Reviewer: Florian Luca (Morelia)
MSC:
11B83 | Special sequences and polynomials |
05A15 | Exact enumeration problems, generating functions |
11Y55 | Calculation of integer sequences |
30B10 | Power series (including lacunary series) in one complex variable |
Keywords:
binomial coefficients; Stirling numbers; Eulerian numbers; MacMahon numbers; Deleham numbersSoftware:
OEISOnline Encyclopedia of Integer Sequences:
Tangent (or ”Zag”) numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).Euler (or secant or ”Zig”) numbers: e.g.f. (even powers only) sec(x) = 1/cos(x).
Octagonal numbers: n*(3*n-2). Also called star numbers.
Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1).
Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
Triangle of Eulerian numbers T(n,k) (n >= 1, 1 <= k <= n) read by rows.
Expansion of E.g.f.: (1 + x)/(1 + x + x^2/2).
Expansion of e.g.f.: 1/2 + exp(-4*x)/2.
Numerator of Bernoulli number B_n.
Denominator of Bernoulli number B_n.
Rhombic matchstick numbers: a(n) = n*(3*n+2).
Closed walks of length 2n along the edges of a cube based at a vertex.
Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).
Triangle read by rows: T(0,0) = 1, T(n,k) = Sum_{j=max(0,1-k)..n-k} (2^j)*(binomial(k+j,1+j) + binomial(k+j+1,1+j))*T(n-1,k-1+j).
T(n, k) = [x^k] (2*n)! [z^(2*n)] 1/cos(z)^x, triangle read by rows, for 0 <= k <= n.
Number of closed walks of length 2*n on the 4-cube.
Number of closed walks of length 2*n on the 5-cube.
Triangle of f-vectors of the simplicial complexes dual to the permutohedra of type B_n.
Square array read by antidiagonals: Hilbert transform of triangle A060187.