The row polynomials with exponents in increasing order (e.g., third row: 1+3x+3x^2) are Grosswald's y_{n}(x) polynomials, p. 18, Eq. (7).
Also called Bessel coefficients.
The triangle a(n,k) has factorization [C(n,k)][C(k,n-k)]Diag((2n-1)!!) The triangle a(n-k,k) is
A100861, which gives coefficients of scaled Hermite polynomials. -
Paul Barry, May 21 2005
Related to k-matchings of the complete graph K_n by a(n,k)=
A100861(n+k,k). Related to the Morgan-Voyce polynomials by a(n,k)=(2k-1)!!*
A085478(n,k). -
Paul Barry, Aug 17 2005
Related to Hermite polynomials by a(n,k)=(-1)^k*
A060821(n+k, n-k)/2^n. -
Paul Barry, Aug 28 2005
The row polynomials, the Bessel polynomials y(n,x):=Sum_{m=0..n} (a(n,m)*x^m) (called y_{n}(x) in the Grosswald reference) satisfy (x^2)*(d^2/dx^2)y(n,x) + 2*(x+1)*(d/dx)y(n,x) - n*(n+1)*y(n,x) = 0.
a(n-1, m-1), n >= m >= 1, enumerates unordered n-vertex forests composed of m plane (aka ordered) increasing (rooted) trees. Proof from the e.g.f. of the first column Y(z):=1-sqrt(1-2*z) (offset 1) and the Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w). See their remark on p. 28 on plane recursive trees. For m=1 see the D. Callan comment on
A001147 from Oct 26 2006. -
Wolfdieter Lang, Sep 14 2007
The asymptotic expansions of the higher order exponential integrals E(x,m,n), see
A163931 for information, lead to the Bessel numbers of the first kind in an intriguing way. For the first four values of m these asymptotic expansions lead to the triangles
A130534 (m=1),
A028421 (m=2),
A163932 (m=3) and
A163934 (m=4). The o.g.f.s. of the right hand columns of these triangles in their turn lead to the triangles
A163936 (m=1),
A163937 (m=2),
A163938 (m=3) and
A163939 (m=4). The row sums of these four triangles lead to
A001147,
A001147 (minus a(0)),
A001879 and
A000457 which are the first four right hand columns of
A001498. We checked this phenomenon for a few more values of m and found that this pattern persists: m = 5 leads to
A001880, m=6 to
A001881, m=7 to
A038121 and m=8 to
A130563 which are the next four right hand columns of
A001498. So one by one all columns of the triangle of coefficients of Bessel polynomials appear. -
Johannes W. Meijer, Oct 07 2009
a(n,k) also appear as coefficients of (n+1)st degree of the differential operator D:=1/t d/dt, namely D^{n+1}= Sum_{k=0..n} a(n,k) (-1)^{n-k} t^{1-(n+k)} (d^{n+1-k}/dt^{n+1-k}. -
Leonid Bedratyuk, Aug 06 2010
a(n-1,k) are the coefficients when expanding (xI)^n in terms of powers of I. Let I(f)(x) := Integral_{a..x} f(t) dt, (xI)^0(f)(x) :=f(x), and for all n>=1, (xI)^n(f)(x) := x*I( (tI)^(n-1)f)(x). Then: (xI)^n = Sum_{k=0..n-1} (-1)^k * a(n-1,k) * x^(n-k) * I^(n+k) where I^(n) denotes iterated integration. -
Abdelhay Benmoussa, Apr 11 2025
a(n,k) counts the number of ways to sequentially place k labeled elements into n ordered steps. At step j, a nonnegative number m(j) of elements is placed, with Sum_{j=1..n} m(j) = k. The term (j - Sum_{r=1..j-1} m(r))_m(j) where (x)_k denotes the falling factorial enumerates arrangements of these m(j) elements given previous placements. The product over all steps gives the total for a fixed composition (m(1),..,m(n)), and summing over all compositions counts all valid placements. -
Abdelhay Benmoussa, Aug 28 2025
