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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
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
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-François Alcover, Sep 11 2012 *)
CROSSREFS
Sequence in context: A011328 A048277 A059419 * A049218 A212358 A344178
KEYWORD
nonn,easy,tabl
AUTHOR
Peter Bala, Jan 27 2011
STATUS
approved

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Last modified March 28 16:28 EDT 2024. Contains 371254 sequences. (Running on oeis4.)