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A298673 Inverse matrix of A135494.

%I A298673 #44 Mar 21 2025 17:17:37
%S A298673 1,1,1,4,3,1,26,19,6,1,236,170,55,10,1,2752,1966,645,125,15,1,39208,
%T A298673 27860,9226,1855,245,21,1,660032,467244,155764,32081,4480,434,28,1,
%U A298673 12818912,9049584,3031876,635124,92001,9576,714,36,1,282137824,198754016,66845340,14180440,2108085,230097,18690,1110,45,1
%N A298673 Inverse matrix of A135494.
%C A298673 Since this is the inverse matrix of A135494 with row polynomials q_n(t), first introduced in that entry by _R. J. Mathar_, and the row polynomials p_n(t) of this entry are a binomial Sheffer polynomial sequence, the row polynomials of the inverse pair are umbral compositional inverses, i.e., p_n(q.(t)) = q_n(p.(t)) = t^n. For example, p_3(q.(t))  = 4q_1(t) + 3q_2(t) + q_3(t) = 4t + 3(-t + t^2) + (-t -3t^2 +t^3) = t^3. In addition, both sequences possess the umbral convolution property (p.x) + p.(y))^n = p_n(x+y) with p_0(t) = 1.
%C A298673 This is the inverse of the Bell matrix generated by A153881; for the definition of the Bell matrix see the link. - _Peter Luschny_, Jan 26 2018
%H A298673 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%H A298673 Andrew Elvey Price and Alan D. Sokal, <a href="https://arxiv.org/abs/2001.01468">Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials</a>, arXiv:2001.01468 [math.CO], 2020.
%F A298673 E.g.f.: e^[p.(t)x] = e^[t*h(x)] = exp[t*[(x-1)/2 + T{ (1/2) * exp[(x-1)/2] }], where T is the tree function of A000169 related to the Lambert function. h(x) = sum(j=1,...) A000311(j) * x^j / j! = exp[xp.'(0)], so the first column of this entry's matrix is A000311(n) for n > 0 and the second column of the full matrix for p_n(t) to n >= 0. The compositional inverse of h(x) is h^(-1)(x) = 1 + 2x - e^x.
%F A298673 The lowering operator is L = h^(-1)(D) = 1 + 2D - e^D with D = d/dt, i.e., L p_n(t) = n * p_(n-1)(t). For example, L p_3(t) = (D - D^2! - D^3/3! - ...) (4t + 6t^ + t^3) = 3 (t + t^2) = 3 p_2(t).
%F A298673 The raising operator is R = t * 1/[d[h^(-1)(D)]/dD] = t * 1/[2 - e^D)] = t (1 + D + 3D^2/2! + 13D^3/3! + ...). The coefficients of R are A000670. For example, R p_2(t) = t (1 + D + 3D^2/2!  + ...)  (t + t^2) = 4t + 3t^2 + t^3   = p_3(t).
%F A298673 The row sums are A006351, or essentially 2*A000311.
%F A298673 Conjectures from _Mikhail Kurkov_, Mar 01 2025: (Start)
%F A298673 T(n,k) = Sum_{j=0..n-k} binomial(n+j-1, k-1)*A269939(n-k, j) for 1 <= k <= n.
%F A298673 T(n,k) = A(n-1,k,0) for n > 0, k > 0 where A(n,k,q) = A(n-1,k,q+1) + 2*(q+1)!*Sum_{j=0..q} A(n-1,k,j)/j! for n >= 0, k > 0, q >= 0 with A(0,k,q) = Stirling1(q+1,k) for k > 0, q >= 0 (see A379458). In other words, T(n,k) = Sum_{j=0}^{n-1} A379460(n-j-1,j)*Stirling1(j+1,k) for n > 0, k > 0.
%F A298673 Recursion for the n-th row (independently of other rows): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} b(j-1)*binomial(-k,j)*T(n,k+j-1)*(-1)^j for 1 <= k < n with T(n,n) = 1 where b(n) = 1 + 4*Sum_{i=1..n} A135148(i).
%F A298673 Recursion for the k-th column (independently of other columns): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} c(j-1)*binomial(n,j)*T(n-j+1,k) for 1 <= k < n with T(n,n) = 1 where c(n) = A000311(n+1) + (n-1)*A000311(n). (End)
%e A298673 Matrix begins as
%e A298673      1;
%e A298673      1;    1;
%e A298673      4,    3,    1;
%e A298673     26,   19,    6,    1;
%e A298673    236,  170,   55,   10,    1;
%e A298673   2752, 1966,  645,  125,   15,    1;
%p A298673 # The function BellMatrix is defined in A264428. Adds (1,0,0,0, ..) as column 0.
%p A298673 BellMatrix(n -> `if`(n=0, 1, -1), 9): MatrixInverse(%); # _Peter Luschny_, Jan 26 2018
%t A298673 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A298673 B = BellMatrix[Function[n, If[n == 0, 1, -1]], rows = 12] // Inverse;
%t A298673 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%Y A298673 Cf. A000311, A000169, A000670, A006351, A135494.
%K A298673 nonn,tabl
%O A298673 1,4
%A A298673 _Tom Copeland_, Jan 24 2018