A128813 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	12	20	8													40
3	144	324	236	56												760
4	2880	8064	8284	3716	616											23560
5	86400	288000	377544	244024	77936	9856										1083760
6	3628800	13996800	22192848	18554976	8641840	2128544	216832									69360640
7	203212800	889056000	1648706688	1682671248	1022037344	369811824	73870368	6288128								5895654400
8	14631321600	71530905600	151601953536	182154477312	135845524944	64441833056	19001703984	3185948832	232660736							642626329600
9	1316818944000	7110822297600	16934597475840	23367592820736	20605203201312	12048659122464	4674477679136	1160813778144	167493112512	10702393856						87397180825600
10	144850083840000	855932313600000	2261011771008000	3518772668928000	3575157550105536	2479243882744512	1188917455562944	389460265627456	83429813952384	10556877624832	599334055936					14507932017049600
11	19120211066880000	122688021012480000	355801018784256000	615965780956032000	707678565432106752	566795748379346496	323046444278190912	131066224585541440	37106573238754240	6983305381287552	786422896247296	40155381747712				2887078471392870400
12	2982752926433280000	20649827952230400000	65197312590329856000	124198942313097216000	159059152902935181312	144326743416314486784	95172109429366155456	45967000133321546688	16142857167503435200	4020814925342443072	674363096936294784	68391648356179456	3172275158069248			678463440777324544000
13	542861032610856960000	4032681956537794560000	13765695063045230592000	28602360259294040064000	40375068521139146870784	40900909368839273275392	30599384310445573077120	17121828091766207799168	7166964016751207501696	2216931175822640677504	492649056773910413312	74488684918963781120	6869385727537113088	291849314542370816		185898982772986925056000
14	114000816848279961600000	904406480329687695360000	3318260250632504647680000	7465659331134542856192000	11510614576924389089648640	12868948230696996956135424	10761367098290533313386752	6839118636178134384000000	3319976221244971602067968	1225253732698382537455616	338451006559721098610944	67863423851016897846272	9338371604192954547200	789443243172831858688	30936027341491306496	58744078556263868317696000

Technical details

Node id: n-96014d301579d4a7 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-bd198274b898c9ae

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 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:             1;
  1:             2,             2;
  2:            12,            20,              8;
  3:           144,           324,            236,             56;
  4:          2880,          8064,           8284,           3716,            616;
  5:         86400,        288000,         377544,         244024,          77936,           9856;
  6:       3628800,      13996800,       22192848,       18554976,        8641840,        2128544,        216832;
  7:     203212800,     889056000,     1648706688,     1682671248,     1022037344,      369811824,      73870368,       6288128;
  8:   14631321600,   71530905600,   151601953536,   182154477312,   135845524944,    64441833056,   19001703984,    3185948832,    232660736;
  9: 1316818944000, 7110822297600, 16934597475840, 23367592820736, 20605203201312, 12048659122464, 4674477679136, 1160813778144, 167493112512, 10702393856;

%F Formula

The row polynomials satisfy 2*P(n,x) = (2*n^2+2*n+(n^2+n+2)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 2*n + (n^2 + n + 2)*x)*P[n - 1, x])/(2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 2*n + (n^2 + n + 2)*x)*v[i - 1])/(2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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 coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A128813 → basis A013609

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.