A185423 - OEIS

%I #22 Oct 29 2023 18:23:00

%S 1,1,2,3,6,6,9,30,36,24,39,150,270,240,120,189,918,1980,2520,1800,720,

%T 1107,6174,16254,25200,25200,15120,5040,7281,47070,142884,266616,

%U 327600,272160,141120,40320,54351,394470,1369710,2948400,4331880

%N Ordered forests of k increasing plane unary-binary trees with n nodes.

%C An increasing tree is a labeled rooted tree with the property that the sequence of labels along any path starting from the root is increasing. A080635 enumerates increasing plane (ordered) unary-binary trees with n nodes with labels drawn from the set {1,2,...,n}. The entry T(n,k) of the present table counts ordered forests of k increasing plane unary-binary trees with n nodes. See below for an example. See A185421 for the corresponding table in the non-plane case. See A185422 for the table for unordered forests of increasing plane unary-binary trees.

%F TABLE ENTRIES

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

%F where P(n,x) are the polynomials described in A185415.

%F Compare (1) with the formula for the number of ordered set partitions

%F (2)... A019538(n,k) = Sum_{j = 0..k} (-1)^j*binomial(k,j)*j^n.

%F RECURRENCE RELATION

%F (3)... T(n+1,k) = k*(T(n,k-1) + T(n,k) + T(n,k+1))

%F GENERATING FUNCTION

%F Let E(t) = 1/2 + (sqrt(3)/2)*tan(sqrt(3)/2*t + Pi/6) be the e.g.f. for A080635.

%F The e.g.f. for the present triangle is

%F (4)... 1/(1 + x*(1 - E(t)) = Sum {n >= 0} R(n,x)*t^n/n!

%F = 1 + x*t + (x + 2*x^2)*t^2/2! + (3*x + 6*x^2 + 6*x^3)*t^3/3! + ....

%F ROW POLYNOMIALS

%F The ordered Bell polynomials OB(n,x) are the row polynomials of A019538 given by

%F (5)... OB(n,x) = Sum_{k = 1..n} k!*Stirling2(n,k)*x^k.

%F By comparing the e.g.f.s for A019538 and the present table we obtain the surprising identity

%F (6)... (-i*sqrt(3))^(n-1)*OB(n,x)/x = R(n,y)/y,

%F where i = sqrt(-1) and x = (sqrt(3)*i/3)*y + (-1/2+sqrt(3)*i/6).

%F RELATIONS WITH OTHER SEQUENCES

%F Column 1: A080635.

%F A185422(n,k) = T(n,k)/k!.

%F Setting y = 0 in (6) gives

%F (7)... A080635(n) = (-i*sqrt(3))^(n-1)*Sum_{k=1..n} k!*Stirling2(n,k) * (-1/2+sqrt(3)*i/6)^(k-1).

%F The row polynomials have their zeros on the vertical line Re(z) = -1/2 in the complex plane (this follows from the above connection with the ordered Bell polynomials, which have negative real zeros). - _Peter Bala_, Oct 23 2023

%e Triangle begins

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

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

%e ..1|....1

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

%e ..3|....3......6......6

%e ..4|....9.....30.....36.....24

%e ..5|...39....150....270....240....120

%e ..6|..189....918...1980...2520...1800....720

%e ..7|.1107...6174..16254..25200..25200..15120...5040

%e ..

%e Examples of recurrence relation:

%e T(5,3) = 270 = 3*(T(4,2) + T(4,3) + T(4,4)) = 3*(30+36+24);

%e T(6,1) = 1*(T(5,0) + T(5,1) + T(5,2)) = 39+150.

%e Examples of ordered forests:

%e T(4,2) = 30. The 15 unordered forests consisting of two plane increasing unary-binary trees on 4 nodes are shown below. We can order the trees in a forest in two ways giving rise to 30 ordered forests.

%e ......... ......... ...3.....

%e .2...3... .3...2... ...|.....

%e ..\./.... ..\./.... ...2.....

%e ...1...4. ...1...4. ...|.....

%e ......... ......... ...1...4.

%e .

%e ......... ......... ...4.....

%e .2...4... .4...2... ...|.....

%e ..\./.... ..\./.... ...2.....

%e ...1...3. ...1...3. ...|.....

%e ......... ......... ...1...3.

%e .

%e ......... ......... ...4.....

%e .3...4... .4...3... ...|.....

%e ..\./.... ..\./.... ...3.....

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

%e ......... ......... ...1...2.

%e .

%e ......... ......... ...4.....

%e .3...4... .4...3... ...|.....

%e ..\./.... ..\./.... ...3.....

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

%e ......... ......... ...2...1.

%e .

%e ......... ......... ..........

%e ..2..4... ..3..4... ..4...3...

%e ..|..|... ..|..|... ..|...|...

%e ..1..3... ..1..2... ..1...2...

%e ......... ......... ..........

%p #A185423

%p P := proc(n,x)

%p description 'polynomial sequence P(n,x) A185415'

%p if n = 0 return 1

%p else

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

%p end proc:

%p with combinat:

%p T:=(n,k) -> add ((-1)^(k-j)*binomial(k,j)*P(n,j),j = 0..k):

%p for n from 1 to 10 do

%p seq(T(n,k),k = 1..n);

%p end do;

%o (PARI) {T(n,k)=if(n<1|k<1|k>n,0,if(n==1,if(k==1,1,0),k*(T(n-1,k-1)+T(n-1,k)+T(n-1,k+1))))}

%Y Cf. A019538, A080635, A185421, A185422.

%K nonn,easy,tabl

%O 1,3

%A _Peter Bala_, Jan 28 2011