A038284 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(38 + 48\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1444 + 3648\,x + 2304\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1444	3648	2304															7396
1	54872	207936	262656	110592														636056
2	2085136	10535424	19961856	16809984	5308416													54700816
3	79235168	500432640	1264250880	1596948480	1008599040	254803968												4704270176
4	3010936384	22819728384	72062300160	121368084480	114980290560	58095304704	12230590464											404567235136
5	114415582592	1011674625024	3833714368512	8070977617920	10194919096320	7726675525632	3253337063424	587068342272										34792782221696
6	4347792138496	43935583715328	194241528004608	490715439169536	774813851320320	782969786597376	494507233640448	178468776050688	28179280429056									2992179271065856
7	165216101262848	1878246203830272	9490086082510848	27970780032663552	52997267430309888	66943916754075648	56373824635011072	30518160704667648	9637313906737152	1352605460594688								257327417311663616
8	6278211847988224	79303728606167040	450779088919265280	1518413773201735680	3356493603919626240	5087737673309749248	5355513340326051840	3865633689257902080	1831089642280058880	513990075025981440	64925062108545024							22130157888803070976
9	238572050223552512	3314895855737782272	20936184352028098560	79337119649790689280	200430618062629109760	354445724573912530944	447720915251257933824	403958720527450767360	255131823491021537280	107423925680430120960	27138675961371820032	3116402981210161152						1903193578437064103936
10	9065737908494995456	137417500928766246912	954690006452481294336	4019747395589394923520	11424545229569859256320	23089607200814873444352	34026789559095602970624	36841035312103509983232	29085027877976455249920	16328436703425378385920	6187618119192774967296	1421079759431833485312	149587343098087735296					163674647745587512938496
11	344498040522809827328	5657020454900877164544	42874260289775069036544	198575521342116109221888	627080593711945608069120	1425783244650318435188736	2401319148884746838212608	3033245240696522321952768	2873600754344073778692096	2016561932873034230661120	1018894450293743611281408	351006700579662870872064	73896147490455341236224	7180192468708211294208				14076019706120526112710656
12	13090925539866773438464	231502683231328203964416	1900758872846694727286784	9603834304909615464185856	33360687585475506349277184	84279631794885489724489728	159687723400835664741138432	230526638292935696468410368	254792600218507875044032512	214562189657690842142343168	135512961889067900300427264	62245188236126882434646016	19656375232461120768835584	3819862393352768408518656	344649238497994142121984			1210537694726365245693116416
13	497455170514937390661632	9425466388704076875694080	83340965963278153427189760	456182129483206734548828160	1728690174883730783553454080	4803939012308472914295914496	10113555815386258766938767360	16425022978371668373374238720	20747397446364212682156933120	20383408017480630003522600960	15448477655353740634248708096	8869939323648080746937057280	3734711294167612946078760960	1088660782105538996427816960	196450065943856661009530880	16543163447903718821855232		104106241746467411129608011776
14	18903296479567620845142016	382045570955471916028133376	3619379093262365520266526720	21335287286599207277360578560	87586968860775693033375006720	265526810862141048353810546688	614904193575484533029877055488	1109601552316663819001281904640	1576802205923680163843926917120	1770444582089746148877391626240	1565445735742512384270535753728	1078584621755606618827546165248	567676116713477167803971665920	220635251840055903276037570560	59720820046932424946897387520	10058243376325461043687981056	794071845499378503449051136	8953136790196197357146289012736

Technical details

Node id: n-9cc88df18a45b6a2 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-f3eacc80e11c3732

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:               1444,                3648,                 2304;
  1:              54872,              207936,               262656,               110592;
  2:            2085136,            10535424,             19961856,             16809984,               5308416;
  3:           79235168,           500432640,           1264250880,           1596948480,            1008599040,             254803968;
  4:         3010936384,         22819728384,          72062300160,         121368084480,          114980290560,           58095304704,           12230590464;
  5:       114415582592,       1011674625024,        3833714368512,        8070977617920,        10194919096320,         7726675525632,         3253337063424,          587068342272;
  6:      4347792138496,      43935583715328,      194241528004608,      490715439169536,       774813851320320,       782969786597376,       494507233640448,       178468776050688,        28179280429056;
  7:    165216101262848,    1878246203830272,     9490086082510848,    27970780032663552,     52997267430309888,     66943916754075648,     56373824635011072,     30518160704667648,      9637313906737152,      1352605460594688;
  8:   6278211847988224,   79303728606167040,   450779088919265280,  1518413773201735680,   3356493603919626240,   5087737673309749248,   5355513340326051840,   3865633689257902080,   1831089642280058880,    513990075025981440,    64925062108545024;
  9: 238572050223552512, 3314895855737782272, 20936184352028098560, 79337119649790689280, 200430618062629109760, 354445724573912530944, 447720915251257933824, 403958720527450767360, 255131823491021537280, 107423925680430120960, 27138675961371820032, 3116402981210161152;

%F Formula

The row polynomials satisfy P(n,x) = (38+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1444 + 3648*x + 2304*x^2.

%t Mathematica

P[0, x_] := 1444 + 3648*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(38 + 48*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] = 1444 + 3648*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (38 + 48*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    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

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
[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.