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A185415 Table of coefficients of a polynomial sequence of binomial type related to A080635. 8
1, 0, 1, 2, 0, 1, 0, 8, 0, 1, 18, 0, 20, 0, 1, 0, 148, 0, 40, 0, 1, 378, 0, 658, 0, 70, 0, 1, 0, 5040, 0, 2128, 0, 112, 0, 1, 14562, 0, 33992, 0, 5628, 0, 168, 0, 1, 0, 277164, 0, 158480, 0, 12936, 0, 240, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Define a sequence of polynomials P(n,x) by means of the recurrence

relation

(1)... P(n+1,x) = x*{P(n,x-1)-P(n,x)+P(n,x+1)}

with starting value P(0,x) = 1. The first few polynomials are

P(1,x) = x

P(2,x) = x^2

P(3,x) = x*(x^2+2),

P(4,x) = x^2*(x^2+8),

P(5,x) = x*(x^4+20*x^2+18).

This triangle lists the coefficients of these polynomials in

ascending powers of x. The triangle has links with A080635, which

gives the number of ordered increasing 0-1-2 trees on n nodes (plane

unary-binary trees in the notation of [BERGERON et al.]). The number of

forests of k such trees on n nodes is given by the formula

... 1/k!*sum {j = 0..k} (-1)^(k-j)*binomial(k,j)*P(n,j).

See A185422.

We also have A080635(n) = P(n,1), which can be used to calculate the terms of A080635 - see A185416.

For similarly defined polynomial sequences for other families of trees see A147309 and A185419. See also A185417.

Exponential Riordan array [(3/2)(1-sqrt(3)*tan((pi+3*sqrt(3)*x)/6))/(-1+2*sin((pi-6*sqrt(3))/6)), log((1/2)(1+sqrt(3)*tan(sqrt(3)*x/2+pi/6))]. Production matrix is the exponential Riordan array [2*cosh(x)-1,x] beheaded. A185422*A008277^{-1}.

REFERENCES

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of increasing trees

FORMULA

GENERATING FUNCTION

The e.g.f. is

(1)... F(x, t) = E(t)^x = Sum_{n >= 0} P(n, x) * t^n/n!,

where

E(t) = 1/2+sqrt(3)/2*tan[sqrt(3)/2*t+Pi/6] = 1 + t + t^2/2! + 3*t^3/3! + 9*t^4/4! + ... is the e.g.f. for A080635.

ROW POLYNOMIALS

One easily checks that

... d/dt(F(x,t)) = x*(F(x-1,t)-F(x,t)+F(x+1,t))

and hence the row generating polynomials P(n,x) satisfy the recurrence

relation

(2)... P(n+1,x) = x*{P(n,x-1)-P(n,x)+P(n,x+1)}.

RELATIONS WITH OTHER SEQUENCES

A080635(n) = P(n,1).

A185422(n,k) = 1/k!*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*P(n,j).

A185423(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*P(n,j).

EXAMPLE

Example

Triangle begins

n\k|....1......2......3......4......5......6......7......8

==========================================================

..1|....1

..2|....0......1

..3|....2......0......1

..4|....0......8......0......1

..5|...18......0.....20......0......1

..6|....0....148......0.....40......0......1..

..7|..378......0....658......0.....70......0......1

..8|....0...5040......0...2128......0....112......0......1

..

MAPLE

#A185415

P := proc(n, x)

description 'polynomial sequence P(n, x)'

if n = 0

return 1

else return

x*(P(n-1, x-1)-P(n-1, x)+P(n-1, x+1))

end proc:

with(PolynomialTools):

for n from 1 to 10

CoefficientList(P(n, x), x);

end do;

MATHEMATICA

p[0][x_] = 1; p[n_][x_] := p[n][x] = x*(p[n-1][x-1] - p[n-1][x] + p[n-1][x+1]) // Expand; row[n_] := CoefficientList[ p[n][x], x]; Table[row[n] // Rest, {n, 1, 10}] // Flatten (* Jean-Fran├žois Alcover, Sep 11 2012 *)

CROSSREFS

Cf. A080635, A147309, A185417, A185419,  A185422, A185423.

Sequence in context: A011328 A048277 A059419 * A049218 A212358 A154469

Adjacent sequences:  A185412 A185413 A185414 * A185416 A185417 A185418

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Jan 27 2011

STATUS

approved

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Last modified February 20 15:54 EST 2018. Contains 299380 sequences. (Running on oeis4.)