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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.

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

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 formulae, including expansion theorems.

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: A254101 A112475 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 August 28 22:18 EDT 2015. Contains 261164 sequences.