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A215534 Matrix inverse of triangle A088956. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

LINKS

Table of n, a(n) for n=0..54.

Eric W. Weisstein,MathWorld: Abel Polynomial

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].

EXAMPLE

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

...

MATHEMATICA

(* The function RiordanArray is defined in A256893. *)

rows = 10;

R = RiordanArray[-#/ProductLog[-#]&, #&, rows, True];

R // Flatten (* Jean-Fran├žois Alcover, Jul 20 2019 *)

CROSSREFS

Cf. A000169, A007318, A088956.

Sequence in context: A098122 A159931 A159755 * A304573 A094322 A136757

Adjacent sequences:  A215531 A215532 A215533 * A215535 A215536 A215537

KEYWORD

sign,easy,tabl

AUTHOR

Peter Bala, Sep 11 2012

STATUS

approved

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Last modified April 5 06:13 EDT 2020. Contains 333238 sequences. (Running on oeis4.)