Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 36\,x + 36\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
19363681
2816481944259212966561
372987484374011664017496013996846656531441
465611049767348322939328734832011757312117573126718464167961643046721
55904911809801062882056687040198404640476171136793618560906992640680244480302330880604661763486784401
6531441127545841403004249353361604209012720134688407043142729497653875362816673442035205986151424035916908544130606940162176782336282429536481
74782969133923132174100071613928005728766040315043064161260169192483780482101139149824367699351219249026580162564902658016256356556946636817827847331845485491486727836416409622876792454961
843046721137749507220662426080192849310080125352051552060168984744962206196107315263034174494720141826892613120252136697978880352991377170432385081502367744320901251973120197477693521920846332972236802256887925964828211099074561853020188851841
9387420489139471376042371013392682529080952192189681071414401062213999920644602927332989441578146514167808433990291396147296442286976921601735961165584588825250344226684928294587349311324162719267839796838419423341712834560103591155801784323884668342566912914039610015744101559956668416150094635296999121
103486784401139471376040264995614476031799473737120270295526765520172989137129932886494568564966403459782742598656011244293913445632029984783769188352065966524292214374411993913507675340801799087026151301120221426095526313984022142609552631398401771408764210511872110713047763156992052100257770897408017366752590299136036561584400629760365615844006297612157665459056928801
113138105960913807666227962899609907871638661465438288036728392166373602644444235978899214985184003880428868503698303453388825688886863795020807992098135402895360207794551520475279364533699305901278822483117820608190111744127873570166446325760164408875928288133120175369467656840675328153448284199735590912108316435905695711232601757977253865062402533717798963642368076011533968909271041447838742264938496131621703842267136984770902183611232881
1228242953648113556617751088311802208275024457309905470035248017540074353696384140320594829568243288869710058726412511999013660163072531759958080556930561890702073175313530885672106219525940592641443808855879330332672312825252107188238745657752354235173213306889075369951241504948224121004932683220065976321361305492686225742233612812286989988006985728996511210332400543334462937550126256876421123146877506312843821056119881047859536907468832694831234419156582456860576059859402752473838133832161689679766443076872509863361
13254186582832913217702307310833044255768277005287080922924320060801430613629680053505258939994118403745368125799588288021402103575997647360010165999198598882496004066399679439552998400138257589100944801945604022038955663848783872010055097389159621959680021657132838189954990080040220389556638487838720064352623290621580541952088484857024604673245184010409983179365255675904001040998317936525567590400876630162472863635865600613641113731004545105920350652064989145454346240159387302267793388339200554390616583629176832001385976541459072942080022175624663345167073281705817281795782082566461081889226673298932241
1422876792454961128110037747781634589710191901032599554976659617888749443720824522360071946597199154146560551590578526848456960346714077931161887232018202489091385999079680808999515172711070208003074198157656302066790401006101215232971585495040285062010982675282556928070169110395735454167859201503623794194331160739840028067644158294181667143680456099217572280452091084806439047777491018147168256078699472836001332909834240828415503536856135892992007455739531831705223036928056805634528241563604090880361490401543355404753305601886036877617506459582464078584869900729435815936002514715836823341946109952580319039266925064486912859731910025074169610246140942214464815497216523347633027360537213511521
Technical details
Node id
n-59c900e21016c37e · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-90334921cba1a1bb

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,          36,           36;
  2:        81,         648,         1944,          2592,           1296;
  3:       729,        8748,        43740,        116640,         174960,          139968,           46656;
  4:      6561,      104976,       734832,       2939328,        7348320,        11757312,        11757312,          6718464,          1679616;
  5:     59049,     1180980,     10628820,      56687040,      198404640,       476171136,       793618560,        906992640,        680244480,        302330880,          60466176;
  6:    531441,    12754584,    140300424,     935336160,     4209012720,     13468840704,     31427294976,      53875362816,      67344203520,      59861514240,       35916908544,       13060694016,        2176782336;
  7:   4782969,   133923132,   1741000716,   13928005728,    76604031504,    306416126016,    919248378048,    2101139149824,    3676993512192,    4902658016256,     4902658016256,     3565569466368,     1782784733184,      548549148672,       78364164096;
  8:  43046721,  1377495072,  20662426080,  192849310080,  1253520515520,   6016898474496,  22061961073152,   63034174494720,  141826892613120,  252136697978880,   352991377170432,   385081502367744,   320901251973120,   197477693521920,    84633297223680,    22568879259648,    2821109907456;
  9: 387420489, 13947137604, 237101339268, 2529080952192, 18968107141440, 106221399992064, 460292733298944, 1578146514167808, 4339902913961472, 9644228697692160, 17359611655845888, 25250344226684928, 29458734931132416, 27192678397968384, 19423341712834560, 10359115580178432, 3884668342566912, 914039610015744, 101559956668416;
%F Formula
The row polynomials satisfy P(n,x) = (9+36*x+36*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 + 36*x + 36*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 + 36*x + 36*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.