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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. 4
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, 70, -21, 1, -1, 111, -70, -91, 70, 154, -28, 1, -1, 237, -883, -1218, -861, -126, 294, -36, 1, -1, 491, -4410, -4495, -3885, -2877, -840, 510, -45, 1 (list; table; graph; refs; listen; history; text; internal format)
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

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

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 .

The e.g.f. is 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) } .

P(n,1) = (-1)^n * A074051(n) and P(n,-1) = A126617(n) .

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)

EXAMPLE

From R. J. Mathar, Mar 22 2013: (Start)

The matrix inverse starts

     1;

     1,    1;

     4,    3,    1;

    26,   19,    6,    1;

   236,  170,   55,   10,    1;

  2752, 1966,  645,  125,   15,    1;

(End)

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.

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-Fran├žois Alcover, Apr 23 2014 *)

CROSSREFS

Cf. A000169, A000311, A008277, A074051, A126617, A134991, A264428.

Cf. A298673 for the inverse matrix.

Sequence in context: A293012 A274391 A126799 * A016566 A096744 A180051

Adjacent sequences:  A135491 A135492 A135493 * A135495 A135496 A135497

KEYWORD

sign,tabl

AUTHOR

Tom Copeland, Feb 08 2008

EXTENSIONS

More terms from Vincenzo Librandi, Jan 21 2018

STATUS

approved

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Last modified February 19 02:06 EST 2018. Contains 299330 sequences. (Running on oeis4.)