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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. 2
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 (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

Table of n, a(n) for n=1..36.

FORMULA

Row polynomials are P(n,t) = sum(j=1,...,n) C(n,j) * x^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.

EXAMPLE

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; - R. J. Mathar, Mar 22 2013

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

Sequence in context: A112475 A274391 A126799 * A016566 A096744 A180051

Adjacent sequences:  A135491 A135492 A135493 * A135495 A135496 A135497

KEYWORD

sign,tabl

AUTHOR

Tom Copeland, Feb 08 2008

STATUS

approved

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Last modified September 28 20:21 EDT 2016. Contains 276605 sequences.