Real-rooted atlas

A299989 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 16\,x + 31\,x^{2} + 26\,x^{3} + 8\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28x29Pn(1)
0011
10310821
2096014816064441
302727011162440297619205129261
40811080626420640422565504044544204804096194481
502434050302401332003832807528001022080947200573440204800327684084101
607291458013316473440027237607156800136604801908480019368960139264006733824196608026214485766121
7021875103055112436514801658059254583200134339520251113600358238720388147200314417152184647680743178241835008020971521801088541
80656117496021811681687392090665568358767360108149529625334016004660568320675573760076906332166803292160458476748822754099207843348481677721601677721637822859361
901968359049083455927383312045818524821179491207557484032212852966404796768332887140108800127913822208151362109440143312289792107100241920617852436482654994432080027320321509949440134217728794280046581
10059049196830031099140309679200217965168011526364800475174252801563743232004172158886409113795328001638884021248243034542080029668685414402965320499200240285646848015543042048007837895884802969567232007952400384013421772800107374182416679880978201
110177147649539011345281212557754009884540160588530448002752531372801036508140800319553030016081572920704001736817450086430970912737280463151320023045800741027840060596722728960524140412928003711726256128021163199692800947845595136032111591424007736309841921181116006408589934592350277500542221
120531441212576404067295124953817440431189969762854507392001493366181120633288015360022153697406720647192443392001592416866078723318528197836805875999193866248849408527564801132385326989312122726863339520011202659824435208539745550336005370054652723202737236344832001102601319874563377991778304073959336837121030792151040687194767367355827511386641
130159432369087330143701646419098808560182149001424132717272688076795579111683621590818560014173682320512046635774831360013021470824693763105729280143360635526144885964811185644310405120169473638636257282208518599213056024692566897197056235827325566976001911288980570112013023021996441600736406046297292833937301490892801242068034781184347291055554560696815494103048933531975680549755813888154472377739119461
140478296922320522050146772767221928896074893122158459557986633603776759301369619606068548352084880601476070431061687424768009703913316185088260695206516326406054025380776755212195538166095872021354062628580556832521435109588992043050847152190259249435535457910784049070653016165580841885982449991680030522551979894374418800597104852992096575989299019776406123947071897601361849948202598435019445344665606484369824808967696581394432043980465111043243919932521508681
Technical details
Node id
n-a58579bc6ff6a601 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-013327fdb00ea32a

Observed on the stored rows ?

  • Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, intercept 1)
  • 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: 0,     1;
  1: 0,     3,     10,       8;
  2: 0,     9,     60,     148,      160,        64;
  3: 0,    27,    270,    1116,     2440,      2976,       1920,        512;
  4: 0,    81,   1080,    6264,    20640,     42256,      55040,      44544,       20480,        4096;
  5: 0,   243,   4050,   30240,   133200,    383280,     752800,    1022080,      947200,      573440,      204800,        32768;
  6: 0,   729,  14580,  133164,   734400,   2723760,    7156800,   13660480,    19084800,    19368960,    13926400,      6733824,      1966080,       262144;
  7: 0,  2187,  51030,  551124,  3651480,  16580592,   54583200,  134339520,   251113600,   358238720,   388147200,    314417152,    184647680,     74317824,     18350080,     2097152;
  8: 0,  6561, 174960, 2181168, 16873920,  90665568,  358767360, 1081495296,  2533401600,  4660568320,  6755737600,   7690633216,   6803292160,   4584767488,   2275409920,   784334848,   167772160,   16777216;
  9: 0, 19683, 590490, 8345592, 73833120, 458185248, 2117949120, 7557484032, 21285296640, 47967683328, 87140108800, 127913822208, 151362109440, 143312289792, 107100241920, 61785243648, 26549944320, 8002732032, 1509949440, 134217728;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3+16*x+31*x^2+26*x^3+8*x^4)*P(n-1,x) for n >= 1, with P(0,x) = x.
%t Mathematica
P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((3 + 16*x + 31*x^2 + 26*x^3 + 8*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = ((3 + 16*x + 31*x^2 + 26*x^3 + 8*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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

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.