Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(9 + 12\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1912425
2812162169616625
372929164860432021605766415625
4656134992816481088649072048384161283072256390625
559049393660118098020995202449440195955210886404147201036801536010249765625
65314414251528155889363464208051963120554273284311014424634368102643203041280608256737284096244140625
74782969446410441934445245158520649457287841260971712126097171296074035256043187224908083283026944201277443354624344064163846103515625
843046721459165024229582512071425670401547556192024760899072302633210882882221056021616657920128098713605977939968217379635260383232012386304017694720157286465536152587890625
938742048946490458682634459325293669664896234174162240437125102848631402926336721603344384661469732352489977579520293986547712142538932224554318069761705594060840609382407219445769024307270778882621443814697265625
10348678440146490458680294439571640117775828656033369818119207118894532096118648242201601581976562688017138079429120152338483814401117148881305667705992806403385299640320138884087808046294695936012345252249625719275520403439616044826624031457280104857695367431640625
113138105960946025554093232217887865241431906127344045343694032560108824865678144205558079614272313231359412224391539199265280406040651089920351901897611264255928652808192156400843382784802055607091203437381173248012221799727104356469158707283875096166415532425216021799895040217998950413841203241943042384185791015625
12282429536481451887258369634644689808336169374039062976592809136720416158082436458777633372958807964165721078652793856810486142479129696057616886415369605761688641536815034325096857658863590145884163622374778208256189743440763289684330418117017631623906793881699212256608256257216961576965415093927936902515654656114605162496104186511366039797761677721659604644775390625
132541865828329440590076910363671583974253001958178119601600750634945847280022018625078186880513767918491027209786055590305280015494588017983360020659450690644480023414044116063744022704527627698176018920439689748480013583905418280960084090843065548800448484496349593602055553941602304080609958494208002686998616473600754245225676800175990552657920335220100300805079092428800588880281600490733568002617245696671088641490116119384765625
14228767924549614270334591592723843301132433448222057398762821449252391615117560029607653168376192075664002541405824015853410056294553602774346759851546880411014334792821760052061815740424089605679470808046264320536394465204369408044011853555230310403143703825373593600195608238023245824010595446226259148804986092341769011202031370954054041607127617382645760021382852147937280543056562487296011519381628518402003370718003202782459330560029679566192642283043553281127428915226843545637252902984619140625
Technical details
Node id
n-23c9ebec99e56671 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6b852c5cd7d4c499

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:         9,         12,           4;
  2:        81,        216,         216,          96,           16;
  3:       729,       2916,        4860,        4320,         2160,          576,           64;
  4:      6561,      34992,       81648,      108864,        90720,        48384,        16128,         3072,          256;
  5:     59049,     393660,     1180980,     2099520,      2449440,      1959552,      1088640,       414720,       103680,        15360,         1024;
  6:    531441,    4251528,    15588936,    34642080,     51963120,     55427328,     43110144,     24634368,     10264320,      3041280,       608256,        73728,        4096;
  7:   4782969,   44641044,   193444524,   515852064,    945728784,   1260971712,   1260971712,    960740352,    560431872,    249080832,     83026944,     20127744,     3354624,      344064,      16384;
  8:  43046721,  459165024,  2295825120,  7142567040,  15475561920,  24760899072,  30263321088,  28822210560,  21616657920,  12809871360,   5977939968,   2173796352,   603832320,   123863040,   17694720,   1572864,    65536;
  9: 387420489, 4649045868, 26344593252, 93669664896, 234174162240, 437125102848, 631402926336, 721603344384, 661469732352, 489977579520, 293986547712, 142538932224, 55431806976, 17055940608, 4060938240, 721944576, 90243072, 7077888, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (9+12*x+4*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 + 12*x + 4*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 + 12*x + 4*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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.