A272867 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 18\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	9	18	6																											33
2	81	324	432	216	36																									1089
3	729	4374	10206	11664	6804	1944	216																							35937
4	6561	52488	174960	314928	332424	209952	77760	15552	1296																					1185921
5	59049	590490	2558790	6298560	9710280	9762768	6473520	2799360	758160	116640	7776																			39135393
6	531441	6377292	34012224	106288200	216119340	300441312	292253184	200294208	96053040	31492800	6718464	839808	46656																	1291467969
7	4782969	66961566	424089918	1607077584	4062335004	7231849128	9334938312	8865853056	6223292208	3214155168	1203654816	317447424	55847232	5878656	279936															42618442977
8	43046721	688747536	5050815264	22499086176	68032951056	147851137728	238561739136	291212661888	271604614752	194141774592	106027439616	43807744512	13438607616	2962842624	443418624	40310784	1679616													1406408618241
9	387420489	6973568802	58113073350	297538935552	1047585002256	2690247875616	5216573837664	7802132087808	9117639881568	8383435008192	6078426587712	3467614261248	1545651507456	531406987776	137953580544	26121388032	3401222400	272097792	10077696											46411484401953
10	3486784401	69735688020	650866421520	3765727153080	15132644300340	44853994534464	101659136313600	180259005121920	253796579540640	286381225468800	260313508726272	190920816979200	112798479795840	53410075591680	20080817049600	5906698868736	1328517469440	220399211520	25395793920	1813985280	60466176									1531578985264449
11	31381059609	690383311398	7133960884446	46025554093200	207882085987620	698667911134776	1813099994244792	3721319466948864	6138786125941920	8227323491682240	9020463114218688	8122217862698496	6013642076145792	3656588218525440	1818899592871680	735075450261504	238762139159808	61337100566016	12166852769280	1795845427200	185570694144	11972302848	362797056							50542106513726817
12	282429536481	6778308875544	76820833922832	546783582627216	2742202512872856	10306042072549344	30141214864554816	70319682565754688	133111425504025584	206871978493787904	266108707633900032	284832237770316288	254445478899197184	189888158513544192	118270536726177792	61295401035196416	26293614914375424	9260205111539712	2646142320070656	603188562087936	106996470552576	14223095783424	1332190789632	78364164096	2176782336					1667889514952984961
13	2541865828329	66088511536554	815091642284166	6344497107509184	34982852106682584	145394725380418800	473232906164118672	1237255263089074944	2644604404907143536	4679771560911079776	6917342534617436064	8594678748305774592	9013641835761868032	7998005473429344768	6009094557174578688	3819857221469233152	2049582973219981056	924399320673793536	348260662374603264	108620489489301504	27697216272979968	5673075704524800	909984264523776	110023286390784	9423290732544	509367066624	13060694016			55040353993448503713
14	22876792454961	640550188738908	8540669183185440	72168654597916968	433937166749013564	1976310848989110816	7086098324964701952	20525858031038323392	48911431816752352848	97144354915062751296	162359597337399221760	229942903723332299136	277331049199065371328	285795674794412375040	252127800550870622208	190530449862941583360	123258244088473498368	68131230732839199744	32071031572819599360	12792672252189333504	4294007731511866368	1201330511190171648	276488518700040192	51408380566093824	7525134358769664	834343255130112	65825897840640	3291294892032	78364164096	1816331681783800622529

Technical details

Node id: n-32697a227a12c599 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-3c4a6b4bc3495597

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:         9,         18,           6;
  2:        81,        324,         432,          216,            36;
  3:       729,       4374,       10206,        11664,          6804,          1944,           216;
  4:      6561,      52488,      174960,       314928,        332424,        209952,         77760,         15552,          1296;
  5:     59049,     590490,     2558790,      6298560,       9710280,       9762768,       6473520,       2799360,        758160,        116640,          7776;
  6:    531441,    6377292,    34012224,    106288200,     216119340,     300441312,     292253184,     200294208,      96053040,      31492800,       6718464,        839808,         46656;
  7:   4782969,   66961566,   424089918,   1607077584,    4062335004,    7231849128,    9334938312,    8865853056,    6223292208,    3214155168,    1203654816,     317447424,      55847232,      5878656,       279936;
  8:  43046721,  688747536,  5050815264,  22499086176,   68032951056,  147851137728,  238561739136,  291212661888,  271604614752,  194141774592,  106027439616,   43807744512,   13438607616,   2962842624,    443418624,    40310784,    1679616;
  9: 387420489, 6973568802, 58113073350, 297538935552, 1047585002256, 2690247875616, 5216573837664, 7802132087808, 9117639881568, 8383435008192, 6078426587712, 3467614261248, 1545651507456, 531406987776, 137953580544, 26121388032, 3401222400, 272097792, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (9+18*x+6*x^2)*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[(9 + 18*x + 6*x^2)*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] = (9 + 18*x + 6*x^2)*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 read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    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 read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.