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%I #46 Jan 08 2021 04:05:04
%S 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 70,-21,1,-1,111,-70,-91,70,154,-28,1,-1,237,-883,-1218,-861,-126,294,
%U -36,1,-1,491,-4410,-4495,-3885,-2877,-840,510,-45,1
%N 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 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 Also the Bell transform of A153881. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%D S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
%D G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.
%H Vincenzo Librandi, <a href="/A135494/b135494.txt">Rows n = 1..25</a>
%H J. Taylor, <a href="https://digital.lib.washington.edu/researchworks/handle/1773/36757">Formal group laws and hypergraph colorings</a>, doctoral thesis, Univ. of Wash., 2016, p. 95.
%F 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 E.g.f.: exp{ t * [ -exp(x) + 2x + 1] } and [ P(.,t) + P(.,s) ]^n = P(n,s+t).
%F 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 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 Therefore P(n+1,t) = 2t * { [ (1+D)/2 * P(n,t) ] - n * P(n-1,t) }.
%F P(n,1) = (-1)^n * A074051(n) and P(n,-1) = A126617(n).
%F See Rota, Roman, Mathworld or Wikipedia on Sheffer sequences and umbral calculus for more formulas, including expansion theorems.
%F From _Tom Copeland_, Jan 20 2018: (Start)
%F 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 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! + ... .
%F (End)
%e The triangle begins:
%e [1] 1;
%e [2] -1, 1;
%e [3] -1, -3, 1;
%e [4] -1, -1, -6, 1;
%e [5] -1, 5, 5, -10, 1;
%e [6] -1, 19, 30, 25, -15, 1;
%e [7] -1, 49, 49, 70, 70, -21, 1.
%e 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.
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> `if`(n=0,1,-1), 9); # _Peter Luschny_, Jan 27 2016
%t 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 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t rows = 12;
%t M = BellMatrix[If[# == 0, 1, -1] &, rows];
%t Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%Y Cf. A000169, A000311, A008277, A074051, A126617, A134991, A264428.
%Y Cf. A298673 for the inverse matrix.
%K sign,tabl
%O 1,5
%A _Tom Copeland_, Feb 08 2008
%E More terms from _Vincenzo Librandi_, Jan 21 2018
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