A200536 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(45 + 114\,x + 72\,x^{2}\right)\,P_{n-1}(x)\)

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

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	45	114	72																											231
2	2025	10260	19476	16416	5184																									53361
3	91125	692550	2191860	3697704	3506976	1772928	373248																							12326391
4	4100625	41553000	184145400	466132320	737166096	745811712	471412224	170201088	26873856																					2847396321
5	184528125	2337356250	13318830000	44960346000	99570027600	151159989024	159312044160	115098485760	54553927680	15318097920	1934917632																			657748550151
6	8303765625	126217237500	879091987500	3709851840000	10565086446000	21390327564480	31570322723136	34224524103168	27046621301760	15195553136640	5761217249280	1323483660288	139314069504																	151939915084881
7	373669453125	6626404968750	54545775637500	276247460475000	961646622930000	2434093927725600	4619848089003840	6679223959722624	7391756942406144	6231280454977536	3938904567521280	1810415356968960	571953912348672	111172627464192	10030613004288															35098120384607511
8	16815125390625	340786541250000	3236874270750000	19126455301800000	78693604371900000	239051758915872000	554618428616851200	1002482523130199040	1426689656224932096	1603972037008318464	1419823177259139072	979156004519411712	515726405611683840	200555419945402368	54305738805215232	9147919059910656	722204136308736													8107665808844335041
9	756680642578125	17252318650781250	185719696914375000	1254230786416500000	5954683048634700000	21105504831340440000	57875669318944512000	125549941045122777600	218416569027377921280	307000104140390920704	349466510443804674048	321407849075514310656	237058741530396721152	138317036462672707584	62439377324051791872	21042500817559486464	4985375152939204608	740981443852763136	51998697814228992											1872870801843041394471
10	34050628916015625	862615932539062500	10378631693601562500	78854597779837500000	424314865017877500000	1718886201576663600000	5439169849627011600000	13767169997246711040000	28308487076340007968000	47754089310687514444800	66449997811946985624576	76406542897100023111680	72469726915430420398080	56390328308722528419840	35646143526515583221760	18023828177044516110336	7118822142415774679040	2116736991272726691840	445758837012478033920	59278515508221050880	3743906242624487424									432633155225742562122801
11	1532278301220703125	42699488660683593750	567828287803476562500	4793729260306078125000	28830854554645275000000	131399304723136197000000	471266340494242352400000	1363347819247101098400000	3234959527294570252320000	6367337785485462253248000	10472427152452471573509120	14451888611300958441252864	16755883443923954517614592	16300384730842783368314880	13250394223798559753502720	8934836268217801758474240	4941585742500906689101824	2204514517589876174487552	773922358924588131287040	205889103988928764968960	39020862813753720176640	4694858428251107229696	269561249468963094528							99938258857146531850367031
12	68952523554931640625	2096156716069921875000	30530338696162265625000	283524604706939062500000	1884757227355780593750000	9544834638512727840000000	38262327588612892116000000	124535764622528983785600000	334926006638010635944800000	753476629444218089245440000	1430052815370912975845222400	2302640003443078171470274560	3155544811641465168888139776	3684224005508925074352439296	3660935207349537218163769344	3086240274203517293549322240	2194971077102866503727841280	1305852139248329517019693056	641935334616917631414829056	256217203863375925953429504	80949706523927142067077120	19483662477242095003238400	3356846239636997416157184	368759789273541513314304	19408409961765342806016					23085737796000848857434784161
13	3102863559971923828125	102187639908408691406250	1617791688655228125000000	16389989106731790468750000	119334064553724862968750000	664793654190531352987500000	2945618410647647321730000000	10653242847088590376056000000	32023635052058911001426400000	81054588134545113346315200000	174363384926268711875040768000	320895138427206299388189388800	507464279623082578408433602560	691312269022930285944967471104	811942847396932125453493764096	821491554373648126433764835328	714192424657996643840166985728	531199348798554854826411294720	335792151483477246622316888064	178731756346060251874612740096	79070842126419646413577125888	28552670033366655139439443968	8200574472912017942925803520	1802098360197357913774227456	284604923679326986907418624	28763263563336238038515712	1397405517247104682033152			5332805430876196086067435141191
14	139628860198736572265625	4952170241715190429687500	84673423115361834960937500	929335272383032003125000000	7354972664668219372031250000	44699877013383234176625000000	216931357704732893852175000000	863061570034536619014840000000	2867617787477380905099132000000	8065184346979624461822825600000	19392277092768476547959468160000	40153637456506164787157864448000	71972102078431581413636037734400	112064429949822130362034602885120	151884454934337943888540730474496	179303087919715408579255364616192	184248581320784848418908256600064	164469299021849250968198612779008	127089227155167487904707170533376	84569587418185066996803311763456	48110643025950114559123638976512	23167632610829677307394684616704	9317130868187054035105310834688	3071752158523250321653021802496	808687796677733808471753621504	163490390094003177010923307008	23833449799407993904416423936	2230259205526379072524910592	100613197241791537106386944	1231878054532401295881577517615121

Technical details

Node id: n-25f8fdf7d4ce854d · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-d4d934822159f131

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:               1;
  1:              45,               114,                 72;
  2:            2025,             10260,              19476,               16416,                5184;
  3:           91125,            692550,            2191860,             3697704,             3506976,              1772928,               373248;
  4:         4100625,          41553000,          184145400,           466132320,           737166096,            745811712,            471412224,             170201088,              26873856;
  5:       184528125,        2337356250,        13318830000,         44960346000,         99570027600,         151159989024,         159312044160,          115098485760,           54553927680,           15318097920,            1934917632;
  6:      8303765625,      126217237500,       879091987500,       3709851840000,      10565086446000,       21390327564480,       31570322723136,        34224524103168,        27046621301760,        15195553136640,         5761217249280,         1323483660288,          139314069504;
  7:    373669453125,     6626404968750,     54545775637500,     276247460475000,     961646622930000,     2434093927725600,     4619848089003840,      6679223959722624,      7391756942406144,      6231280454977536,      3938904567521280,      1810415356968960,       571953912348672,       111172627464192,       10030613004288;
  8:  16815125390625,   340786541250000,   3236874270750000,   19126455301800000,   78693604371900000,   239051758915872000,   554618428616851200,   1002482523130199040,   1426689656224932096,   1603972037008318464,   1419823177259139072,    979156004519411712,    515726405611683840,    200555419945402368,    54305738805215232,     9147919059910656,     722204136308736;
  9: 756680642578125, 17252318650781250, 185719696914375000, 1254230786416500000, 5954683048634700000, 21105504831340440000, 57875669318944512000, 125549941045122777600, 218416569027377921280, 307000104140390920704, 349466510443804674048, 321407849075514310656, 237058741530396721152, 138317036462672707584, 62439377324051791872, 21042500817559486464, 4985375152939204608, 740981443852763136, 51998697814228992;

%F Formula

The row polynomials satisfy P(n,x) = (45+114*x+72*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[(45 + 114*x + 72*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] = (45 + 114*x + 72*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 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_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    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

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.