A215534 Matrix inverse of triangle A088956.

1, -1, 1, -1, -2, 1, -4, -3, -3, 1, -27, -16, -6, -4, 1, -256, -135, -40, -10, -5, 1, -3125, -1536, -405, -80, -15, -6, 1, -46656, -21875, -5376, -945, -140, -21, -7, 1, -823543, -373248, -87500, -14336, -1890, -224, -28, -8, 1, -16777216, -7411887, -1679616, -262500, -32256, -3402, -336, -36, -9, 1

Offset: 0

Author

Peter Bala, Sep 11 2012

Keywords

• sign
• easy
• tabl

Comments

For commuting lower unitriangular matrices A and B, we define A raised to the matrix power B, denoted A^^B, to be the matrix Exp(B*log(A)). Here Exp is the matrix exponential and the matrix logarithm Log(A) is defined as sum {n >= 1} (-1)^(n+1)*(A-1)^n/n. This triangle, call it M, is related to Pascal's triangle P by M^^M = P^(-1). Also M = P^(-1)^^A088956.

Examples

			Triangle begins
.n\k.|......0......1.....2......3......4......5......6
= = = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|......1
..1..|.....-1......1
..2..|.....-1.....-2.....1
..3..|.....-4.....-3....-3......1
..4..|....-27....-16....-6.....-4......1
..5..|...-256...-135...-40....-10.....-5......1
..6..|..-3125..-1536..-405....-80....-15.....-6......1
...
		

Links

• T. Copeland, Composition, Conjugation, and the Umbral Calculus-Part I, 2021.
• Eric Weisstein's World of Mathematics, Abel Polynomial.

Crossrefs

Cf. A000169, A007318, A088956.

Programs

Mathematica

(* The function RiordanArray is defined in A256893. *)
rows = 10;
R = RiordanArray[-#/ProductLog[-#]&, #&, rows, True];
R // Flatten (* Jean-François Alcover, Jul 20 2019 *)

Formula

T(n,k) = -binomial(n,k)*(n-k-1)^(n-k-1) for n,k >= 0.

E.g.f.: (x/T(x))*exp(t*x) = exp(-T(x))*exp(t*x) = 1 + (-1 + t)*x + (-1 - 2*t + t^2)*x^2/2! + ...., where T(x) := sum {n >= 0} n^(n-1) *x^n/n! denotes the tree function of A000169. The triangle is the exponential Riordan array [x/T(x),x] belonging to the exponential Appell group.

Let A(n,x) = x*(x+n)^(n-1) be an Abel polynomial. This is the triangle of connection constants expressing A(n,x) as a linear combination of the basis polynomials A(k,x+1), 0 <= k <= n. For example, A(4,x) = -27*A(0,x+1) - 16*A(1,x+1) - 6*A(2,x+1) - 4*A(3,x+1) + A(4,x+1) giving row 4 as [-27,-16,-6,-4,1].