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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 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-François 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.)