Real-rooted atlas

A034870 → x ↦ x+1 → basis A013610

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 18\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1918936
281324486324811296
37294374109351458010935437472946656
46561524881837083674164592703674161837085248865611679616
55904959049026572057085880124002901488034812400290708588026572055904905904960466176
65314416377292350751061169170202630632954209012724910514844209012722630632951169170203507510663772925314412176782336
747829696696156643525017917410007164787751969957550393814363255907164151496081436325590795755039384787751969174100071643525017966961566478296978364164096
8430467216887475365165606520241061637607834503222018802807732834471814176849245448824055401129927049245448824034471814176818802807732878345032220241061637605165606520688747536430467212821109907456
93874204896973568802592753348173161351190241185506696340331941874975271920739577961232926964193616952745757662188363841751801695274575766212329269641936719207395779633194187497521185506696340316135119024592753348176973568802387420489101559956668416
1034867844016973568802066248903619039749342171401689347042284554059105353104135147763382760270295526765520439230230993970585640307991960644204338791156585640307991960439230230993970270295526765520135147763382760540591053531041689347042284539749342171406624890361906973568802034867844013656158440062976
113138105960969038331139872490247696794832683179786022955245103983582638882374340623414350006063175351851429957296100347214311699301560956667070878020292436671921414221372036420960882029243667192141415609566670708780100347214311699305351851429957296234143500060631782638882374340622955245103983548326831797860724902476967969038331139831381059609131621703842267136
12282429536481677830887554477950552068756571637381837544300109625464710612004385018588424380138858921966769774999229422002420771873362521755136927774866705342455391662300058013670498479290982926476373352565231503670498479290982926455391662300058013636927774866705342420771873362521755197749992294220024380138858921966761200438501858842430010962546471065716373818375447795055206875667783088755442824295364814738381338321616896
1325418658283296608851153655482610639420692566088511536554003800089413351855016720393418748162058521376965618567016720393418748162003971093436952688475794218687390537695013501717685639140815196388620882023866402454857761025298330026436929734118597400245485776102529833001963886208820238664013501717685639140815794218687390537695039710934369526884751672039341874816200585213769656185670167203934187481620380008941335185506608851153655400826106394206925660885115365542541865828329170581728179578208256
1422876792454961640550188738908864742754797525874944372082452236468402325515326475224833116247356708086186027894820071402708703733837202244071103473013226558905158007717807170130900300214663833623248710491260359000474406980695952175250672076555856556523385442555760917739132198688452600856556523385442555760695952175250672076555491260359000474406980300214663833623248710158007717807170130900711034730132265589052708703733837202244086186027894820071402248331162473567080468402325515326475749443720824522368647427547975258640550188738908228767924549616140942214464815497216
Technical details
Node id
n-9335dcc335de4078 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-2cd9f46ca2d53115

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: yes
  • γ-positive (gamma_positive): yes
  • 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,         18,           9;
  2:        81,        324,         486,          324,            81;
  3:       729,       4374,       10935,        14580,         10935,          4374,           729;
  4:      6561,      52488,      183708,       367416,        459270,        367416,        183708,          52488,           6561;
  5:     59049,     590490,     2657205,      7085880,      12400290,      14880348,      12400290,        7085880,        2657205,         590490,          59049;
  6:    531441,    6377292,    35075106,    116917020,     263063295,     420901272,     491051484,      420901272,      263063295,      116917020,       35075106,        6377292,        531441;
  7:   4782969,   66961566,   435250179,   1741000716,    4787751969,    9575503938,   14363255907,    16415149608,    14363255907,     9575503938,     4787751969,     1741000716,     435250179,      66961566,       4782969;
  8:  43046721,  688747536,  5165606520,  24106163760,   78345032220,  188028077328,  344718141768,   492454488240,   554011299270,   492454488240,   344718141768,   188028077328,   78345032220,   24106163760,    5165606520,    688747536,    43046721;
  9: 387420489, 6973568802, 59275334817, 316135119024, 1185506696340, 3319418749752, 7192073957796, 12329269641936, 16952745757662, 18836384175180, 16952745757662, 12329269641936, 7192073957796, 3319418749752, 1185506696340, 316135119024, 59275334817, 6973568802, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (9+18*x+9*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 + 18*x + 9*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 + 18*x + 9*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.