Real-rooted atlas

A038319 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(21 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 441 + 210\,x + 25\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
044121025676
192616615157512517576
21944811852206615010500625456976
340841014862025231525055125065625312511881376
4857661211225230307293037523152500413437539375015625308915776
5180108854130018142352144153025850854375202584375289406252296875781258031810176
6378228593617204354164060036284700285887070008508543750162067500019293750013125000390625208827064576
779428004658117020286712451620979686900900544270500321622953750765768937501215506250012403125007382812519531255429503678976
8166798809782013971400232905042550716781125270163281150001125680338125032162295375006381407812508682187500077519531254101562509765625141167095653376
93502775005422219173934538010551092135064048875780096474320625371474511581250123824837193750294821040937505013963281250596900390625473730468752255859375488281253670344486987776
10735582751138664121016650032533260275218036140316502184270128097750011701447114809375445769413897500012382483719375002527037493750003760472460937539793359375002842382812501230468750024414062595428956661682176
11154472377739119461478128788240131665683041126057330950596305744970685750354943895815884375152118812492521875482916865055625001149802059656250020532179636718752715896777343752586568359375016795898437506665039062512207031252481152873203736576
12324391993252150868110813066441738362270167345075884046082751593762627467105550010435350536987000625496921454142238125017747194790794218754829168650556250001006076802199218751596947305078125019011277441406251645998046875009797607421875358886718750610351562564509974703297150976
136812231858295168230124329399493911315107540548999156518858512541836268971011520687529883049265008229062515653025805480500937562115181767779765625190148515615652343754527345609896484375838397335166015625119771047880859375129622346191406251028748779296875565246582031251922607421875305175781251677259342285725925376
14143056869024198532832154497854866361345840809731759797564526043000108130664417383622700008367253794202304137500478128788240131665000020870701073974001250007098877916317687500001901485156156523437504024307208796875000067071786813281250008710621664062500008641489746093750063307617187500003229980468750001025390625000015258789062543608742899428874059776
Technical details
Node id
n-eadfce40222b7cb9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f594353b5992a359

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:             441,             210,               25;
  1:            9261,            6615,             1575,             125;
  2:          194481,          185220,            66150,           10500,             625;
  3:         4084101,         4862025,          2315250,          551250,           65625,            3125;
  4:        85766121,       122523030,         72930375,        23152500,         4134375,          393750,          15625;
  5:      1801088541,      3001814235,       2144153025,       850854375,       202584375,        28940625,        2296875,         78125;
  6:     37822859361,     72043541640,      60036284700,     28588707000,      8508543750,      1620675000,      192937500,      13125000,       390625;
  7:    794280046581,   1702028671245,    1620979686900,    900544270500,    321622953750,     76576893750,    12155062500,    1240312500,     73828125,     1953125;
  8:  16679880978201,  39714002329050,   42550716781125,  27016328115000,  11256803381250,   3216229537500,   638140781250,   86821875000,   7751953125,   410156250,    9765625;
  9: 350277500542221, 917393453801055, 1092135064048875, 780096474320625, 371474511581250, 123824837193750, 29482104093750, 5013963281250, 596900390625, 47373046875, 2255859375, 48828125;
%F Formula
The row polynomials satisfy P(n,x) = (21+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 210*x + 25*x^2.
%t Mathematica
P[0, x_] := 441 + 210*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 5*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] = 441 + 210*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 5*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.