Real-rooted atlas

A038226 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 729 + 864\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
07298642561849
1196833499220736409679507
253144112597121119744442368655363418801
31434890742515280503884802985984088473601048576147008443
4387420489137749507220407334401612431360716636160169869312167772166321363049
51046035320343391094768771397240327618738176045148078080160526499843170893824268435456271818611107
62824295364811338925209984277703006515232912948920322437996216320115579079884834245653299257982058496429496729611688200277601
77625597484987406698532532649640261511884813329744312729611848661611315270214291030016277389791723527044820107264104367705292868719476736502592611936843
820589113209464912200955975979203253588260261120514147280633856053318977250918403791571715620864187238109413376063403380965376014089640214528018554258718720109951162777621611482313284249
95559060566555523362368392486582241073684125886169601908771779353190402262248034788966401876827999232327681112194369915453444707701036679168013948743812382720275530741972992032655495344947217592186044416929293739471222707
101500946352969991211067339628778660416347873656787118950468715784056714854409162104540895313920868703245358963097660058495975434485763050590271768100864112984824880300032029757320133083136052901902458814464569986827839078428147497671065639959630797262576401
114052555153018976267312196841417758171681110033213929806832642411924020390691389443573220770949172428803811435489012450590723011504583911072071681784595308984339005447931534706597062246426112048416780451840618952258768129228810003268828575825929879771682544025645035996273704961718264124282290785243
12109418989131512359209907772354276250683808349660462387889152281682882479973425576837121350677451418787178086416008029053852292481024142293591589798155386889636814668515430629376499686686515614921523219740708603086021591045849098845358821212161260411872400554065921867276848000820838417023606591460474887205759403792793673885357344138503765449
13295431270655083369864326260557391562966210160108932682513150082056960279728369910311321825280497294879840553461022720648325176681017845481472640321162154091699240960487863742593593675612160289104440055462918881280133249783070830645739520473777006474064518184961276167020805560991744025208237448011081318403447280334770746163202918332558536081408011529215046068469763177070365797955661914307
147976644307687250986336175630405287701342685260833613513461200596749004809295588907788807002193920179026156742599245968179202546149784783633720436326427661874205056761407209472234174596444924964293836801561163976299499761958912082234151837998341370675203411194446613264530931712110260830597600469686681627224896443851967823872049640836820698744750080630359832643793584128049806208999015789363218446744073709551616136614025729312093462315201
Technical details
Node id
n-c16b8d4cf8f42711 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-50db7ebeb5f2c85c

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:              729,               864,                256;
  1:            19683,             34992,              20736,               4096;
  2:           531441,           1259712,            1119744,             442368,              65536;
  3:         14348907,          42515280,           50388480,           29859840,            8847360,            1048576;
  4:        387420489,        1377495072,         2040733440,         1612431360,          716636160,          169869312,           16777216;
  5:      10460353203,       43391094768,        77139724032,        76187381760,        45148078080,        16052649984,         3170893824,         268435456;
  6:     282429536481,     1338925209984,      2777030065152,      3291294892032,      2437996216320,      1155790798848,       342456532992,       57982058496,        4294967296;
  7:    7625597484987,    40669853253264,     96402615118848,    133297443127296,    118486616113152,     70214291030016,     27738979172352,     7044820107264,     1043677052928,      68719476736;
  8:  205891132094649,  1220095597597920,   3253588260261120,   5141472806338560,   5331897725091840,   3791571715620864,   1872381094133760,   634033809653760,   140896402145280,   18554258718720,   1099511627776;
  9: 5559060566555523, 36236839248658224, 107368412588616960, 190877177935319040, 226224803478896640, 187682799923232768, 111219436991545344, 47077010366791680, 13948743812382720, 2755307419729920, 326554953449472, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = (27+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 729 + 864*x + 256*x^2.
%t Mathematica
P[0, x_] := 729 + 864*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 16*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] = 729 + 864*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 16*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

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.