A109692 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-3 + 6\,n + 2\,n x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	27	30	8													65
3	405	612	300	48												1365
4	8505	16092	11196	3408	384											39585
5	229635	519534	463212	203976	44448	3840										1464645
6	7577955	19900242	21520404	12289752	3914496	660096	46080									65909025
7	295540245	882200808	1117899144	780585984	324721872	80546688	11038464	645120								3493178325
8	13299311025	44427680280	64420674408	53012755584	27101859984	8820150912	1785477888	205645824	10321920							213083877825
9	678264862275	2505199292730	4085152639848	3863222674128	2336424459696	937661176224	249822088704	42626539008	4228042752	185794560						14702787569925
10	38661097149675	156361656931110	282957686325936	301906745222256	210440647685232	100175176238688	32993082580608	7426154497536	1093529217024	95151144960	3715891200					1132114642884225
11	2435649120429525	10701328523952780	21266290691018388	25245194048172720	19899709199059248	10940730352112448	4282418079829440	1193695550118144	232267739618304	30052164907008	2327426334720	81749606400				96229744645159125
12	168059789309637225	796847247043050420	1724205942255135492	2252309365908358992	1978964591891233392	1232503415073180864	558064375958930112	185143026874058496	44675167236498432	7648025129422848	881844374863872	61498954874880	1961990553600			8949366251999798625
13	12604484198222791875	64133098050279349350	150033474092254472820	213752556941760447192	206982387905459838192	143890835519660632992	73899916988822460864	28395400790486570112	8164356241462903296	1735156232855672832	264986981479784448	27540375362076672	1746122118266880	51011754393600		903885991451979661125
14	1020963220056046141875	5547706499622865469850	13948438146880434080220	21514894386865721461512	22750645014711539414928	17450664538445386741728	10014836670645117053760	4369225139716441083264	1456384077692119130112	369149629622270791680	70048320019821379584	9650405885762174976	912566401717764096	53023371417354240	1428329123020800	98523573068265783062625

Technical details

Node id: n-58093636829549bd · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-6f5c8093b03ddefa

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:            3,             2;
  2:           27,            30,             8;
  3:          405,           612,           300,            48;
  4:         8505,         16092,         11196,          3408,           384;
  5:       229635,        519534,        463212,        203976,         44448,         3840;
  6:      7577955,      19900242,      21520404,      12289752,       3914496,       660096,        46080;
  7:    295540245,     882200808,    1117899144,     780585984,     324721872,     80546688,     11038464,      645120;
  8:  13299311025,   44427680280,   64420674408,   53012755584,   27101859984,   8820150912,   1785477888,   205645824,   10321920;
  9: 678264862275, 2505199292730, 4085152639848, 3863222674128, 2336424459696, 937661176224, 249822088704, 42626539008, 4228042752, 185794560;

%F Formula

The row polynomials satisfy P(n,x) = (6*n-3+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[(6*n - 3 + 2*n*x)*P[n - 1, x]];
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] = (6*n - 3 + 2*n*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 - 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_A013610 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
    
    ↑ Liu–Wang theorem (above)
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A109692 → basis A013610

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.