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A135494 Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with lowering operator (D-1)/2 + T{ (1/2) * exp[(D-1)/2] } where T(x) is Cayley's Tree function.

%I A135494 #55 Feb 18 2025 08:58:09
%S A135494 1,-1,1,-1,-3,1,-1,-1,-6,1,-1,5,5,-10,1,-1,19,30,25,-15,1,-1,49,49,70,
%T A135494 70,-21,1,-1,111,-70,-91,70,154,-28,1,-1,237,-883,-1218,-861,-126,294,
%U A135494 -36,1,-1,491,-4410,-4495,-3885,-2877,-840,510,-45,1
%N A135494 Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with lowering operator (D-1)/2 + T{ (1/2) * exp[(D-1)/2] } where T(x) is Cayley's Tree function.
%C A135494 The lowering (or delta) operator for these polynomials is L = (D-1)/2 + T{ (1/2) * exp[(D-1)/2] } and the raising operator is R = 2t * { 1 - T[ (1/2) * exp[(D-1)/2] ] }, where T(x) is the tree function of A000169. In addition, L = E(D,1) = A(D) where E(x,t) is the e.g.f. of A134991 and A(x) is the e.g.f. of A000311, so L = sum(j=1,...) A000311(j) * D^j / j! also. The polynomials and operators can be generalized through A134991.
%C A135494 Also the Bell transform of A153881. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%C A135494 Exponential Riordan array [2 - exp(x), 1 + 2*x - exp(x)] belonging to the derivative subgroup of the exponential Riordan group. See the example section for a factorization of this array as an infinite product of arrays. - _Peter Bala_, Feb 13 2025
%D A135494 S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
%D A135494 G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.
%H A135494 Vincenzo Librandi, <a href="/A135494/b135494.txt">Rows n = 1..25</a>
%H A135494 Peter Bala, <a href="/A033184/a033184.pdf">Factorisations of some Riordan arrays as infinite products</a>
%H A135494 J. Taylor, <a href="http://hdl.handle.net/1773/36757">Formal group laws and hypergraph colorings</a>, doctoral thesis, Univ. of Wash., 2016, p. 95.
%F A135494 Row polynomials are P(n,t) = Sum_{j=1..n} C(n,j) * t^j = [ Bell(.,-t) + 2t ]^n, umbrally, where Bell(j,t) are the Touchard/Bell/exponential polynomials described in A008277, with P(0,t) = 1.
%F A135494 E.g.f.: exp{ t * [ -exp(x) + 2x + 1] } and [ P(.,t) + P(.,s) ]^n = P(n,s+t).
%F A135494 The lowering operator gives L[P(n,t)] = n * P(n-1,t) = (D-1)/2 * P(n,t) + Sum_{j>=1} j^(j-1) * 2^(-j) / j! * exp(-j/2) * P(n,t + j/2).
%F A135494 The raising operator gives R[P(n,t)] = P(n+1,t) = 2t * { P(n,t) - Sum_{j>=1} j^(j-1) * 2^(-j) / j! * exp(-j/2) * P(n,t + j/2) } .
%F A135494 Therefore P(n+1,t) = 2t * { [ (1+D)/2 * P(n,t) ] - n * P(n-1,t) }.
%F A135494 P(n,1) = (-1)^n * A074051(n) and P(n,-1) = A126617(n).
%F A135494 See Rota, Roman, Mathworld or Wikipedia on Sheffer sequences and umbral calculus for more formulas, including expansion theorems.
%F A135494 From _Tom Copeland_, Jan 20 2018: (Start)
%F A135494 Define Q(n,z;w) = [Bell(.,w)+z]^n. Then Q(n,z;w) are a sequence of Appell polynomials with e.g.f. exp[(exp(t)-1+z)*w], lowering operator D = d/dz, and raising operator R = z + w*exp(D), and exp[(exp(D)-1)w] z^n = exp[Bell(.,w)D] z^n = Q(n,z;w) = e^(-w) (w d/dw + z)^n e^w =  e^(-w) exp(a.w) = exp[(a. - 1)w] with (a.)^k = a_k = (k + z)^n and (a. - 1)^m = sum{k = 0,..,m} (-1)^k a^(m-k). Then P(n,t) = Q(n,2t;-t).
%F A135494 For example, exp[(a. - 1)w] = (a. - 1)^0 + (a. - 1)^1 w + (a. - 1)^2 w^2/2! + ... = a_0 + (a_1 - a_0) w + (a_2 - 2a_1 + a_0) w^2/2! + ... = z^n + [(1+z)^n - z^n] w + [(2+z)^n - 2(1+z)^n + z^n] w^2/2! + ... . (End)
%F A135494 T(n+1, k) = Sum_{i = 0..n} s(n,k)*binomial(n, i)*T(i, k-1), where s(n,i) = 1 if i = n else -1. - _Peter Bala_, Feb 13 2025
%e A135494 The triangle begins:
%e A135494   [1]  1;
%e A135494   [2] -1,  1;
%e A135494   [3] -1, -3,  1;
%e A135494   [4] -1, -1, -6,   1;
%e A135494   [5] -1,  5,  5, -10,   1;
%e A135494   [6] -1, 19, 30,  25, -15,   1;
%e A135494   [7] -1, 49, 49,  70,  70, -21, 1.
%e A135494 P(3,t) = [B(.,-t) + 2t]^3 = B(3,-t) + 3B(2,-t)2t + 3B(1,-t)(2t)^2 + (2t)^3 = (-t + 3t^2 - t^3) + 3(-t + t^2)(2t) + 3(-t)(2t)^2 + (2t)^3 = -t - 3t + t^3.
%e A135494 From _Peter Bala_, Feb 13 2025: (Start)
%e A135494 The array factorizes as an infinite product of lower triangular arrays:
%e A135494   /  1               \    / 1             \ / 1             \ / 1             \
%e A135494   | -1   1           |   | -1  1          | | 0 -1          | | 0  1          |
%e A135494   | -1  -3   1       | = | -1 -2   1      | | 0 -1  1       | | 0  0  1       | ...
%e A135494   | -1  -1  -6   1   |   | -1 -3  -3  1   | | 0 -1 -2  1    | | 0  0 -1  1    |
%e A135494   | -1   5   5 -10  1|   | -1 -4  -6 -4  1| | 0 -1 -3 -3  1 | | 0  0 -1 -2  1 |
%e A135494   |...               |   |...             | |...            | |...            |
%e A135494 where the first array in the product on the right-hand side is A154926. (End)
%p A135494 # The function BellMatrix is defined in A264428.
%p A135494 # Adds (1,0,0,0, ..) as column 0.
%p A135494 BellMatrix(n -> `if`(n=0,1,-1), 9); # _Peter Luschny_, Jan 27 2016
%t A135494 max = 8; s = Series[Exp[t*(-Exp[x]+2*x+1)], {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]*n!; Table[t[n, k], {n, 0, max}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 23 2014 *)
%t A135494 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A135494 rows = 12;
%t A135494 M = BellMatrix[If[# == 0, 1, -1] &, rows];
%t A135494 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%Y A135494 Cf. A000169, A000311, A008277, A074051, A126617, A134991, A154926, A264428.
%Y A135494 Cf. A298673 for the inverse matrix.
%K A135494 sign,tabl
%O A135494 1,5
%A A135494 _Tom Copeland_, Feb 08 2008
%E A135494 More terms from _Vincenzo Librandi_, Jan 21 2018