

A135494


Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomialtype) with lowering operator (D1)/2 + T{ (1/2) * exp[(D1)/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 = (D1)/2 + T{ (1/2) * exp[(D1)/2] } and the raising operator is R = 2t * { 1  T[ (1/2) * exp[(D1)/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(n1,t) = (D1)/2 * P(n,t) + sum(j=1,...) j^(j1) * 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^(j1) * 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(n1,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 (* JeanFrançois Alcover, Apr 23 2014 *)


CROSSREFS

Sequence in context: A046643 A112475 A126799 * A016566 A096744 A180051
Adjacent sequences: A135491 A135492 A135493 * A135495 A135496 A135497


KEYWORD

sign,tabl


AUTHOR

Tom Copeland, Feb 08 2008


STATUS

approved



