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A185415 Table of coefficients of a polynomial sequence of binomial type related to A080635.

%I A185415 #30 May 11 2025 22:05:42
%S A185415 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 A185415 5040,0,2128,0,112,0,1,14562,0,33992,0,5628,0,168,0,1,0,277164,0,
%U A185415 158480,0,12936,0,240,0,1
%N A185415 Table of coefficients of a polynomial sequence of binomial type related to A080635.
%C A185415 Define a sequence of polynomials P(n,x) by means of the recurrence relation
%C A185415 (1)... P(n+1,x) = x*{P(n,x-1)-P(n,x)+P(n,x+1)}
%C A185415 with starting value P(0,x) = 1. The first few polynomials are
%C A185415   P(1,x) = x
%C A185415   P(2,x) = x^2
%C A185415   P(3,x) = x*(x^2+2),
%C A185415   P(4,x) = x^2*(x^2+8),
%C A185415   P(5,x) = x*(x^4+20*x^2+18).
%C A185415 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
%C A185415 ... 1/k!*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*P(n,j).
%C A185415 See A185422.
%C A185415 We also have A080635(n) = P(n,1), which can be used to calculate the terms of A080635 - see A185416.
%C A185415 For similarly defined polynomial sequences for other families of trees see A147309 and A185419. See also A185417.
%C A185415 Exponential Riordan array [(3/2)*(1-sqrt(3)*tan((Pi+3*sqrt(3)*x)/6))/(-1+2*sin((Pi-6*sqrt(3)*x)/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 A185415 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 A185415 G. C. Greubel, <a href="/A185415/b185415.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A185415 F. Bergeron, Ph. Flajolet and B. Salvy, <a href="http://algo.inria.fr/flajolet/Publications/BeFlSa92.pdf">Varieties of increasing trees</a>
%F A185415 GENERATING FUNCTION
%F A185415 The e.g.f. is
%F A185415 (1)... F(x, t) = E(t)^x = Sum_{n >= 0} P(n, x) * t^n/n!,
%F A185415 where
%F A185415 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 A185415 ROW POLYNOMIALS
%F A185415 One easily checks that
%F A185415 ... d/dt(F(x,t)) = x*(F(x-1,t)-F(x,t)+F(x+1,t))
%F A185415 and hence the row generating polynomials P(n,x) satisfy the recurrence relation
%F A185415 (2)... P(n+1,x) = x*{P(n,x-1)-P(n,x)+P(n,x+1)}.
%F A185415 RELATIONS WITH OTHER SEQUENCES
%F A185415 A080635(n) = P(n,1).
%F A185415 A185422(n,k) = 1/k!*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*P(n,j).
%F A185415 A185423(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*P(n,j).
%e A185415 Triangle begins:
%e A185415   n\k|....1......2......3......4......5......6......7......8
%e A185415   ==========================================================
%e A185415   ..1|....1
%e A185415   ..2|....0......1
%e A185415   ..3|....2......0......1
%e A185415   ..4|....0......8......0......1
%e A185415   ..5|...18......0.....20......0......1
%e A185415   ..6|....0....148......0.....40......0......1..
%e A185415   ..7|..378......0....658......0.....70......0......1
%e A185415   ..8|....0...5040......0...2128......0....112......0......1
%p A185415 #A185415
%p A185415 P := proc(n,x)
%p A185415 description 'polynomial sequence P(n,x)'
%p A185415 if n = 0
%p A185415 return 1
%p A185415 else return
%p A185415 x*(P(n-1,x-1)-P(n-1,x)+P(n-1,x+1))
%p A185415 end proc:
%p A185415 with(PolynomialTools):
%p A185415 for n from 1 to 10
%p A185415 CoefficientList(P(n,x),x);
%p A185415 end do;
%t A185415 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 A185415 Cf. A080635, A147309, A185417, A185419, A185422, A185423.
%K A185415 nonn,easy,tabl
%O A185415 1,4
%A A185415 _Peter Bala_, Jan 27 2011