A161198 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(5 + 18\,x + 2\,n\right)\,P_{n-1}(x)\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	9	18														27
3	99	360	324													783
4	1287	6462	10692	5832												24273
5	19305	120096	276696	279936	104976											801009
6	328185	2389122	6865560	9739440	6823440	1889568										28035315
7	6235515	51300648	173449836	308629440	304955280	158723712	34012224									1037306655
8	130945815	1189552878	4565858220	9603315288	11959390800	8822392992	3571283520	612220032								40454959545
9	3011753745	29716740864	126426690864	303061699584	447925663584	418184073216	240942594816	78364164096	11019960576							1658653341345
10	75293843625	797130089010	3695568607152	9852222925152	16653252182112	18517263774912	13550878188288	6296070809088	1686053968128	198359290368						71322093677835
11	2032933777875	22877801588520	114128693995284	332530253907840	626977821569760	799724661200640	699184459032192	413909719234560	158852731703040	35704672266240	3570467226624					3209494215502575
12	58955079558375	700049054068830	3721532554456596	11697693855242472	24167901395864160	34477615963074240	34671393213545088	24588702120381696	12057104165610240	3894784666375680	746227650364416	64268410079232				150846228128621025
13	1827607466309625	22762712108184480	127968392161393416	429616095492735360	959763432666153456	1503828319980856320	1695410276955234048	1386334843575644160	816366867300787968	337766199638630400	93239181156059136	15424418419015680	1156831381426176			7391465178302430225
14	60311046388217625	784066433963661090	4632685759273303368	16480762210165348368	39405282996852300528	66902076347359020768	83017448899178137344	76266434823190470144	51894133805287597824	25840888199488986624	9156684571645298688	2187311068636581888	315814967129346048	20822964865671168		376964724093423941475
15	2110886623587616875	28527924023716055400	176257197385911517500	660215021022706653504	1675838624672806789104	3050867766100907136384	4109848085723697180864	4163639298996872927232	3189090510002494386432	1838525495477291292672	785619947598387213312	241376209691895742464	50425123084985585664	6413473178626719744	374813367582081024	19979130376951468898175

Technical details

Node id: n-2912565edb620442 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-37b5047703982ae7

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:           1;
   2:           9,           18;
   3:          99,          360,           324;
   4:        1287,         6462,         10692,          5832;
   5:       19305,       120096,        276696,        279936,         104976;
   6:      328185,      2389122,       6865560,       9739440,        6823440,        1889568;
   7:     6235515,     51300648,     173449836,     308629440,      304955280,      158723712,       34012224;
   8:   130945815,   1189552878,    4565858220,    9603315288,    11959390800,     8822392992,     3571283520,     612220032;
   9:  3011753745,  29716740864,  126426690864,  303061699584,   447925663584,   418184073216,   240942594816,   78364164096,   11019960576;
  10: 75293843625, 797130089010, 3695568607152, 9852222925152, 16653252182112, 18517263774912, 13550878188288, 6296070809088, 1686053968128, 198359290368;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+5+18*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 5 + 18*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n + 5 + 18*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.