OFFSET
1,5
COMMENTS
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.
Also the Bell transform of A153881. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
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
REFERENCES
S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.
LINKS
Vincenzo Librandi, Rows n = 1..25
Tomasz Kania and Tom Copeland, Closed form for A135494, answers to question on MathOverflow, 2026.
Tomasz Kania, Identities with Stirling numbers of both kinds, answer to question on MathOverflow, 2026.
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 95.
FORMULA
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.
E.g.f.: exp{ t * [ -exp(x) + 2x + 1] } and [ P(.,t) + P(.,s) ]^n = P(n,s+t).
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).
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) } .
Therefore P(n+1,t) = 2t * { [ (1+D)/2 * P(n,t) ] - n * P(n-1,t) }.
See Rota, Roman, Mathworld or Wikipedia on Sheffer sequences and umbral calculus for more formulas, including expansion theorems.
From Tom Copeland, Jan 20 2018: (Start)
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).
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)
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
T(n,k) = (-1)^k * Sum_{j=0..k} (-1)^j * 2^j * binomial(n,j) * Stirling2(n-j,k-j) (due to Tom Copeland). - Mikhail Kurkov, Apr 09 2026
EXAMPLE
The triangle begins:
[1] 1;
[2] -1, 1;
[3] -1, -3, 1;
[4] -1, -1, -6, 1;
[5] -1, 5, 5, -10, 1;
[6] -1, 19, 30, 25, -15, 1;
[7] -1, 49, 49, 70, 70, -21, 1.
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.
From Peter Bala, Feb 13 2025: (Start)
The array factorizes as an infinite product of lower triangular arrays:
/ 1 \ / 1 \ / 1 \ / 1 \
| -1 1 | | -1 1 | | 0 -1 | | 0 1 |
| -1 -3 1 | = | -1 -2 1 | | 0 -1 1 | | 0 0 1 | ...
| -1 -1 -6 1 | | -1 -3 -3 1 | | 0 -1 -2 1 | | 0 0 -1 1 |
| -1 5 5 -10 1| | -1 -4 -6 -4 1| | 0 -1 -3 -3 1 | | 0 0 -1 -2 1 |
|... | |... | |... | |... |
where the first array in the product on the right-hand side is A154926. (End)
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n=0, 1, -1), 9); # Peter Luschny, Jan 27 2016
MATHEMATICA
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 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 1, -1] &, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)
PROG
(PARI) T(n, k) = my(A = 1); (-1)^k * (stirling(n, k, 2) + sum(j=1, k, (A *= -2*(n-j+1)/j) * stirling(n-j, k-j, 2))) \\ Mikhail Kurkov, Apr 09 2026
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Tom Copeland, Feb 08 2008
EXTENSIONS
More terms from Vincenzo Librandi, Jan 21 2018
STATUS
approved