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A194649 Triangle of coefficients of a sequence of polynomials related to the enumeration of linear labeled rooted trees. 4
1, 1, 3, 4, 13, 36, 24, 75, 316, 432, 192, 541, 3060, 6360, 5760, 1920, 4683, 33244, 92880, 127680, 86400, 23040, 47293, 403956, 1418424, 2620800, 2688000, 1451520, 322560, 545835, 5449756, 23051952, 53548992, 73785600, 60318720, 27095040, 5160960, 7087261, 80985780, 400813080, 1122145920, 1943867520, 2133734400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Define the sequence of polynomials {P(n,x)}n>=0 recursively by setting P(0,x) = 1, P(1,x) = 1 and P(n+1,x) = d/dx((1+x)*(1+2*x)*P(n,x)) for n>=1. The first few values are P(2,x) = 3+4*x, P(3,x) = 13+36*x+24*x^2 and P(4,x) = 75+316*x+432*x^2+192*x^3.

This triangle shows the coefficients of the P(n,x) in ascending powers of x. The values of P(n,x) at an integer or half integer value of x enumerate linear labeled rooted trees: in particular we have P(n,0) = A000670(n), P(n,1/2) = A050351(n), P(n,1) = A050352(n) and P(n,3/2) = A050353(n).

More generally, for m >= 2, P(n,m/2-1), n = 0,1,2,... counts m level linear labeled rooted trees (see the e.g.f. below and the comment of Benoit Cloitre in A050351).

LINKS

Table of n, a(n) for n=0..42.

L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207v5 [math.CO], 2005-2006.

FORMULA

T(n,k) = 2^k*sum {i = k+1..n} Stirling2(n,i)*i!*binomial(i-1,k).

Recurrence: T(n+1,k) = (k+1)*(2*T(n,k-1)+3*T(n,k)+T(n,k+1)).

E.g.f.: G(x,t) := 1 + (1-exp(t))/((2*x+1)*exp(t)-2*x-2) = sum {n>=0} P(n,x)*t^n/n! = 1 + t + (3+4*x)*t^2/2! + (13+36*x+24*x^2)*t^3/3! + ....

Column k generating function: 2^k*((exp(x)-1)/(2-exp(x)))^(k+1) (apart from initial term 1 when k = 0).

The generating function G(x,t) satisfies the partial differential equation d/dx((1+x)*(1+2*x)*G(x,t)) - d/dt(G(x,t)) = 2*(2x+1). Hence the row polynomials P(n,x) satisfy the defining recurrence P(n+1,x) = d/dx((1+x)*(2+x)*P(n,x)), with P(0,x) = P(1,x) = 1.

Reflection property: P(n,x) = (-1)^n*P(n,-x-3/2).

The polynomial P(n,x) has all real zeros, lying in the interval [-1,-1/2] (apply [Liu et al, Theorem 1.1, Corollary 1.2] with f(x) = P(n,x-1/2) and g(x) = P'(n,x-1/2) and use the reflection property).

Row sums are A050352; Column 0: A000670; Column 1: 4*A083411; Main diagonal: A002866.

EXAMPLE

Triangle begins

n\k|......0.......1........2........3........4........5.......6

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

.0.|......1

.1.|......1

.2.|......3.......4

.3.|.....13......36.......24

.4.|.....75.....316......432......192

.5.|....541....3060.....6360.....5760.....1920

.6.|...4683...33244....92880...127680....86400....23040

.7.|..47293..403956..1418424..2620800..2688000..1451520..322560

..

CROSSREFS

A000670, A002866 (main diagonal), A050351, A050352, A050353, A083411 (1/4*column 1).

Sequence in context: A174684 A286917 A084315 * A062165 A243764 A201821

Adjacent sequences:  A194646 A194647 A194648 * A194650 A194651 A194652

KEYWORD

nonn,easy,tabf

AUTHOR

Peter Bala, Sep 01 2011

STATUS

approved

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Last modified February 23 18:13 EST 2019. Contains 320437 sequences. (Running on oeis4.)