Real-rooted atlas

A094638 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + n + 5\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1257
26334584
3242106095851428
41201458661513278994531416
57201126870308218583338508208845848232
6504096876773856328778178341319924615522112527143424
7403209211689000252487440721580186973065867193295828351514126251004306688
83628809621072111400812735704964303072264979738974721308360724212238947180499661662542180880896
93628800109637280146998778411478879684575283101581918757127334258702788846064829397545028072119101848748151251982501402112
103991680013547908802066499410418653717566811034454787824469293556541125514772897772413199377153830396679860924226487186546857579867420125103090072909824
11479001600180537465603089455188483168370842696216355186504441032865692236823517359375216698544004032984211450699878050298163963521759780011101507485023253410940339056255876134155859968
12622702080025816978368049009253264645632715137860043651191374787624028658137796229633609079742115283422661814412546072471817627650392399551853236002947740853928224255883160911738705016713607661375625364320317663318016
138717829120039444090739208229595310553610483291616029929096505815536064567752528215521362622224152467095769073730077665096512352286162084257504451200378928795868762240134437210220565846669053463007225335206579048077240088582120605290812524409461283442307072
14130767436800064135298707200145927061379648020415806751060432196202349444411504137012962380883768871695256395291153922855727278556009059787004553873954912667201760368078122058033350544417645228975999442468802094043365567383541122904602997825204355068148197648675504918087450157631251757481212407846109184
Technical details
Node id
n-b42b8599e76cc9eb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9766abf8728d27af

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:       1;
  1:       2,         5;
  2:       6,        33,         45;
  3:      24,       210,        609,         585;
  4:     120,      1458,       6615,       13278,        9945;
  5:     720,     11268,      70308,      218583,      338508,       208845;
  6:    5040,     96876,     773856,     3287781,     7834131,      9924615,      5221125;
  7:   40320,    921168,    9000252,    48744072,   158018697,    306586719,    329582835,    151412625;
  8:  362880,   9621072,  111400812,   735704964,  3030722649,   7973897472,  13083607242,  12238947180,   4996616625;
  9: 3628800, 109637280, 1469987784, 11478879684, 57528310158, 191875712733, 425870278884, 606482939754, 502807211910, 184874815125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+1+(5*n+2)*x+(4*n+1)*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[Together[((n + 1 + (5*n + 2)*x + (4*n + 1)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 + (5*n + 2)*x + (4*n + 1)*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

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.