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Table of coefficients of a polynomial sequence of binomial type related to A080635.
8

%I #25 Jun 30 2017 21:19:17

%S 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,

%T 5040,0,2128,0,112,0,1,14562,0,33992,0,5628,0,168,0,1,0,277164,0,

%U 158480,0,12936,0,240,0,1

%N Table of coefficients of a polynomial sequence of binomial type related to A080635.

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

%C relation

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

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

%C P(1,x) = x

%C P(2,x) = x^2

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

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

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

%C This triangle lists the coefficients of these polynomials in

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

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

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

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

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

%C See A185422.

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

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

%C 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}.

%D 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.

%H G. C. Greubel, <a href="/A185415/b185415.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H F. Bergeron, Ph. Flajolet and B. Salvy, <a href="http://algo.inria.fr/flajolet/Publications/BeFlSa92.pdf">Varieties of increasing trees</a>

%F GENERATING FUNCTION

%F The e.g.f. is

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

%F where

%F 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.

%F ROW POLYNOMIALS

%F One easily checks that

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

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

%F relation

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

%F RELATIONS WITH OTHER SEQUENCES

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

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

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

%e Example

%e Triangle begins

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

%e ==========================================================

%e ..1|....1

%e ..2|....0......1

%e ..3|....2......0......1

%e ..4|....0......8......0......1

%e ..5|...18......0.....20......0......1

%e ..6|....0....148......0.....40......0......1..

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

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

%e ..

%p #A185415

%p P := proc(n,x)

%p description 'polynomial sequence P(n,x)'

%p if n = 0

%p return 1

%p else return

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

%p end proc:

%p with(PolynomialTools):

%p for n from 1 to 10

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

%p end do;

%t 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 *)

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

%K nonn,easy,tabl

%O 1,4

%A _Peter Bala_, Jan 27 2011