A038329 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(24 + 27\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 576 + 1296\,x + 729\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	576	1296	729															2601
1	13824	46656	52488	19683														132651
2	331776	1492992	2519424	1889568	531441													6765201
3	7962624	44789760	100776960	113374080	63772920	14348907												345025251
4	191102976	1289945088	3627970560	5441955840	4591650240	2066242608	387420489											17596287801
5	4586471424	36118462464	121899810816	228562145280	257132413440	173564379072	65086642152	10460353203										897410677851
6	110075314176	990677827584	3900793946112	8776786378752	12342355845120	11108120260608	6248317646592	2008387814976	282429536481									45767944570401
7	2641807540224	26748301344768	120367356051456	315964309635072	533189772509184	599838494072832	449878870554624	216905884017408	61004779879896	7625597484987								2334165173090451
8	63403380965376	713288035860480	3611020681543680	10833062044631040	21327590900367360	28792247715495936	26992732233277440	17352470721392640	7320573585587520	1830143396396880	205891132094649							119042423827613001
9	1521681143169024	18830804146716672	105923273325281280	357491047472824320	804354856813854720	1266858899481821184	1425216261917048832	1145263067611914240	644210475531701760	241578928324388160	54355258872987336	5559060566555523						6071163615208263051
10	36520347436056576	493024690386763776	3050590271768100864	11439713519130378240	28956774845298769920	52122194721537785856	68410380572018343936	65967152694446260224	46383154238282526720	23191577119141263360	7827157277710176384	1601009443167990624	150094635296999121					309629344375621415601
11	876488338465357824	12818641950055858176	86525833162877042688	356919061796867801088	1003834861303690690560	2032765594139973648384	3049148391209960472576	3430291940111205531648	2894308824468829667328	1808943015293018542080	814024356881858343936	249757473134206537344	46829526212663725752	4052555153018976267				15791096563156692195651
12	21035720123168587776	331312591939905257472	2422723328560557195264	10902254978522507378688	33728851339804007202816	75889915514559016206336	128064232430818339848192	164654013125337865519104	162081294170254461370368	121560970627690846027776	68378045978076100890624	27972836991031132182528	7867360403727505926336	1361658531414376025712	109418989131512359209			805345924720991301978201
13	504857282956046106624	8519466649883278049280	67090799867830814638080	327067649355675221360640	1103853316575403872092160	2732036958524124583428096	5122569297232733593927680	7409430590640203948359680	8335609414470229441904640	7293658237661450761666560	4923219310421479264124928	2517555329192801896427520	944083248447300711160320	245098535654587684628160	39390836087344449315240	2954312706550833698643		41072642160770556400888251
14	12116574790945106558976	218098346237011918061568	1840204796374788058644480	9661075180967637307883520	35323306130412923906949120	95372926552114894548762624	196706661013736970006822912	316135705200648701796679680	400109251894571013211422720	400109251894571013211422720	315086035866974672903995392	193348249282007185645633536	90631991850940868271390720	31372612563787223632404480	7563040528770134268526080	1134456079315520140278912	79766443076872509863361	2094704750199298376445300801

Technical details

Node id: n-c67f40b153a1102f · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-cb7716efe3e0f2cd

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:              576,              1296,                729;
  1:            13824,             46656,              52488,              19683;
  2:           331776,           1492992,            2519424,            1889568,             531441;
  3:          7962624,          44789760,          100776960,          113374080,           63772920,            14348907;
  4:        191102976,        1289945088,         3627970560,         5441955840,         4591650240,          2066242608,           387420489;
  5:       4586471424,       36118462464,       121899810816,       228562145280,       257132413440,        173564379072,         65086642152,         10460353203;
  6:     110075314176,      990677827584,      3900793946112,      8776786378752,     12342355845120,      11108120260608,       6248317646592,       2008387814976,       282429536481;
  7:    2641807540224,    26748301344768,    120367356051456,    315964309635072,    533189772509184,     599838494072832,     449878870554624,     216905884017408,     61004779879896,      7625597484987;
  8:   63403380965376,   713288035860480,   3611020681543680,  10833062044631040,  21327590900367360,   28792247715495936,   26992732233277440,   17352470721392640,   7320573585587520,   1830143396396880,   205891132094649;
  9: 1521681143169024, 18830804146716672, 105923273325281280, 357491047472824320, 804354856813854720, 1266858899481821184, 1425216261917048832, 1145263067611914240, 644210475531701760, 241578928324388160, 54355258872987336, 5559060566555523;

%F Formula

The row polynomials satisfy P(n,x) = (24+27*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 1296*x + 729*x^2.

%t Mathematica

P[0, x_] := 576 + 1296*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 27*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] = 576 + 1296*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 27*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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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
    
    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

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.