A128813 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(2 + 3\,n + 2\,n x + 3\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	40	44	12													96
3	760	1316	756	144												2976
4	23560	55996	49756	19584	2880											151776
5	1083760	3282616	3968656	2393544	720000	86400										11534976
6	69360640	255605344	391863856	319870368	146608848	35769600	3628800									1222707456
7	5895654400	25610650080	47622327024	49133357216	30374492688	11250511488	2311545600	203212800								172401751296
8	642626329600	3216047975520	7034800451376	8784343482272	6848421422544	3413269225728	1061995297536	188581478400	14631321600							31204716984576
9	87397180825600	495218894334720	1246177179183936	1827802754212832	1721976226870464	1080562542727968	451625590780416	121226657840640	18962192793600	1316818944000						7052266038514176
10	14507932017049600	91820026350379520	261339490121352576	440494746909563072	486906356623908544	368790767048593728	193831727769625536	69802440187392000	16482656366208000	2304433152000000	144850083840000					1946425426629912576
11	2887078471392870400	20187232269976071680	64126802012399259264	122155267331021591360	155039671560220125760	137661001717026079680	87252895076569853760	39476473662881578752	12493970721611136000	2634292837587456000	333010342748160000	19120211066880000				644266816214501062656
12	678463440777324544000	5194383824981664627200	18219006707030093109120	38710268936724358414784	55490544520291097805760	56536524166895468343360	41979546610849984063680	22888422942722068193280	9094413010988143245312	2568118249404389376000	489207113209460736000	56442863069429760000	2982752926433280000			251908325139869915498496
13	185898982772986925056000	1546741514266449174860800	5937385693872908474049280	13922472909341951151510656	22249678145043594030268928	25590286724422338126728960	21792043169747870871939840	13911705369480543784548480	6657562140586167660392448	2358847568336644759670784	601440270410991108096000	104501039085145608192000	11089875380478935040000	542861032610856960000		114870196263780681467314176
14	58744078556263868317696000	527809104890525193517772800	2201029597259793404520340480	5646352435065367343427716096	9954617604795585455382219008	12758963015376613594402826240	12260245853769018202146071040	8972427962402904719024686080	5025247764016143175439194368	2143483881117474952738381824	685413114800568589689200640	159324785137214144888832000	25449618828111921192960000	2500417916205607157760000	114000816848279961600000	60421723234748638451807256576

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(1 + \frac{3}{2}\,n + n x + \frac{3}{2}\,n^{2} + n^{2} x\right)\,P_{n-1}(x)\)

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

Technical details

Node id: n-34a4e524e6a5040f · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-8f4e693782942de7)
Fingerprint: f6-52cac31f6d7fd0d8

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:              4,               2;
  2:             40,              44,               12;
  3:            760,            1316,              756,              144;
  4:          23560,           55996,            49756,            19584,             2880;
  5:        1083760,         3282616,          3968656,          2393544,           720000,            86400;
  6:       69360640,       255605344,        391863856,        319870368,        146608848,         35769600,         3628800;
  7:     5895654400,     25610650080,      47622327024,      49133357216,      30374492688,      11250511488,      2311545600,       203212800;
  8:   642626329600,   3216047975520,    7034800451376,    8784343482272,    6848421422544,    3413269225728,   1061995297536,    188581478400,    14631321600;
  9: 87397180825600, 495218894334720, 1246177179183936, 1827802754212832, 1721976226870464, 1080562542727968, 451625590780416, 121226657840640, 18962192793600, 1316818944000;

%F Formula

The row polynomials satisfy 2*P(n,x) = (3*n^2+3*n+2+(2*n^2+2*n)*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[((3*n^2 + 3*n + 2 + (2*n^2 + 2*n)*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] = ((3*n^2 + 3*n + 2 + (2*n^2 + 2*n)*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 (−∞, -1]
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), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (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)
Ore-algebra identity of two definitions polytool ? step › direct check
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) (above)
  - ↑ Geometric basis map (elementary) (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.