Real-rooted atlas

A038321 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 29\,x + 18\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 396\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0121396324841
11331653410692583224389
21464195832235224256608104976707281
31610511317690431244070567205773680188956820511149
41771561173935087115526015524784019053144012471148834012224594823321
5194871712232166861095791004298852092048903069604801392288261894124861222003217249876309
6214358881280615262416071601392525979681921075867531201408408404481152334149125387536281611019960576500246412961
72357947691347261387222272983625448678664751682130217711776348581080108838027026920962666830459392109097609702419835929036814507145975869
82593742460142443058438031253524849801363790175264039053991382560766878376239361045743240326409778378351104060003685336320218195219404803570467226624420707233300201
9285311670611513561007099842018627853620206273264008680675076136755680154653805874937625306986415898882957959451208960242014864189824013200810773990404320265344215046426841007923212200509765705829
1031384283767216162732085197655464588766778430253412054606401113875625646872029163289107845376556753701149775367809012951191654479864905182641920580835674055577602851375127181926484834301304586241156831381426176353814783205469041
1134522712143931734392240152714721039653968119243262379238087152176982460519447440521293792802736096113736827520596966418611480867006776322284136288223558912207648753474868992013591554772900515846065652543277916161654268875439431682082296486567116810260628712958602189
1237974983358324186997234602706129253342225924196460567330933322012827255298919994905768919916010180150976218943392977149160324094525790741490790458626164731071345408639558160702596495365232748587566698598431137016388826636288127378703408836239363206736589313359872374813367582081024297558232675799463481
134177248169415651102532455067475070117446266713653262083280080033317767604088294837999235864014717861416797249110440139622045810679392084449594434043247552013819024543752531417601758784941932140362240172680703389701053747212844019260390987468807005828687485993164802645557686183521894406184420565104336896067466406164774584328629188747598184440949
1445949729863572161120304747279170748814764673529716410080112748416045107131520599616576239887926720235485782668755985766470645734800626795729921651458735599067952128030401853996255569118720442208785400081005363205065300632764564243251245210947796576275890176308256462249383699251201552060509227666178048054422900972918164684801187408748500032684032121439531096594251776250246473680347348787521
Technical details
Node id
n-1ab9a5b82290fb1b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5e3baf07c55bade9

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:          121,           396,            324;
  1:         1331,          6534,          10692,            5832;
  2:        14641,         95832,         235224,          256608,          104976;
  3:       161051,       1317690,        4312440,         7056720,         5773680,          1889568;
  4:      1771561,      17393508,       71155260,       155247840,       190531440,        124711488,         34012224;
  5:     19487171,     223216686,     1095791004,      2988520920,      4890306960,       4801392288,       2618941248,        612220032;
  6:    214358881,    2806152624,    16071601392,     52597968192,    107586753120,     140840840448,     115233414912,      53875362816,      11019960576;
  7:   2357947691,   34726138722,   227298362544,    867866475168,   2130217711776,    3485810801088,    3802702692096,    2666830459392,    1090976097024,     198359290368;
  8:  25937424601,  424430584380,  3125352484980,  13637901752640,  39053991382560,   76687837623936,  104574324032640,   97783783511040,   60003685336320,   21819521940480,   3570467226624;
  9: 285311670611, 5135610070998, 42018627853620, 206273264008680, 675076136755680, 1546538058749376, 2530698641589888, 2957959451208960, 2420148641898240, 1320081077399040, 432026534421504, 64268410079232;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+29*x+18*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 396*x + 324*x^2.
%t Mathematica
P[0, x_] := 121 + 396*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((11 + 29*x + 18*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 396*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 29*x + 18*x^2)*v[i - 1])/(1 + x)); 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.