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A215534 Matrix inverse of triangle A088956.

%I A215534 #18 Feb 16 2025 08:33:18
%S A215534 1,-1,1,-1,-2,1,-4,-3,-3,1,-27,-16,-6,-4,1,-256,-135,-40,-10,-5,1,
%T A215534 -3125,-1536,-405,-80,-15,-6,1,-46656,-21875,-5376,-945,-140,-21,-7,1,
%U A215534 -823543,-373248,-87500,-14336,-1890,-224,-28,-8,1,-16777216,-7411887,-1679616,-262500,-32256,-3402,-336,-36,-9,1
%N A215534 Matrix inverse of triangle A088956.
%C A215534 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.
%H A215534 T. Copeland, <a href="https://tcjpn.wordpress.com/2021/11/20/composition-conjugation-and-the-umbral-calculus-part-i/">Composition, Conjugation, and the Umbral Calculus-Part I</a>, 2021.
%H A215534 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbelPolynomial.html">Abel Polynomial</a>.
%F A215534 T(n,k) = -binomial(n,k)*(n-k-1)^(n-k-1) for n,k >= 0.
%F A215534 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.
%F A215534 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].
%e A215534 Triangle begins
%e A215534 .n\k.|......0......1.....2......3......4......5......6
%e A215534 = = = = = = = = = = = = = = = = = = = = = = = = = = = =
%e A215534 ..0..|......1
%e A215534 ..1..|.....-1......1
%e A215534 ..2..|.....-1.....-2.....1
%e A215534 ..3..|.....-4.....-3....-3......1
%e A215534 ..4..|....-27....-16....-6.....-4......1
%e A215534 ..5..|...-256...-135...-40....-10.....-5......1
%e A215534 ..6..|..-3125..-1536..-405....-80....-15.....-6......1
%e A215534 ...
%t A215534 (* The function RiordanArray is defined in A256893. *)
%t A215534 rows = 10;
%t A215534 R = RiordanArray[-#/ProductLog[-#]&, #&, rows, True];
%t A215534 R // Flatten (* _Jean-François Alcover_, Jul 20 2019 *)
%Y A215534 Cf. A000169, A007318, A088956.
%K A215534 sign,easy,tabl
%O A215534 0,5
%A A215534 _Peter Bala_, Sep 11 2012