Real-rooted atlas

A038323 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(20 + 67\,x + 47\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1204767
2400188022094489
3800056400132540103823300763
4160000150400053016008305840487968120151121
53200000376000001767200004152920004879681002293450071350125107
66400000090240000053016000001661168000029278086000275214008401077921532990458382169
71280000000210560000001484448000005814088000001366310680000192649805880015090901460605066231204636060711605323
825600000000481280000000395852800000018605081600000546524272000001027465631360001207272116848008105969927408023811286661761406067677556641
95120000000001082880000000010179072000000055815244800000019674873792000004623595341120000724363270108800072953729346672004286031599116980111913047310276727206534396294947
101024000000000024064000000000025447680000000001594721280000000065582912640000000184943813644800000362181635054400000486358195644480000428603159911698000223826094620553400525991322358300491822837804551761449
112048000000000005294080000000000622054400000000004385483520000000002061177254400000000678127316697600000015935991942393600000267497007604464000003143089839352452000024620870408260874000115718090918826107802472159215084012303122130132904968017083
1240960000000000001155072000000000001492930560000000000116946227200000000006183531763200000000023250079429632000000063743967769574400000012839856365014272000001885853903611471200000196966963266086992000013886170910259132936005933182116201629527201161914831089485782418182718904632857144561
13819200000000000002502656000000000000352874496000000000003040601907200000000001786353620480000000000755627581463040000000023676330885841920000000556393775817285120000009806440298779650240000012802852612295654480000012034681455557915211200077131367510621183853600302097856083266303426605460999706120583177327548242166610401428685587
14163840000000000000053903360000000000000823373824000000000000773971394560000000000050017901373440000000000235084136455168000000000828671581004467200000000222557510326914048000000045763388060971701120000007169597462885566508800000842427701889054064784000071989276343246438263360004229369985165728247972400152907991771376328965156025666698618766740933436936732225162896895721934329
Technical details
Node id
n-ec8f65a85acdc7d1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7bc4cf22d8541c8f

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:           20,             47;
  2:          400,           1880,            2209;
  3:         8000,          56400,          132540,          103823;
  4:       160000,        1504000,         5301600,         8305840,          4879681;
  5:      3200000,       37600000,       176720000,       415292000,        487968100,        229345007;
  6:     64000000,      902400000,      5301600000,     16611680000,      29278086000,      27521400840,      10779215329;
  7:   1280000000,    21056000000,    148444800000,    581408800000,    1366310680000,    1926498058800,    1509090146060,     506623120463;
  8:  25600000000,   481280000000,   3958528000000,  18605081600000,   54652427200000,  102746563136000,  120727211684800,   81059699274080,   23811286661761;
  9: 512000000000, 10828800000000, 101790720000000, 558152448000000, 1967487379200000, 4623595341120000, 7243632701088000, 7295372934667200, 4286031599116980, 1119130473102767;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (20+67*x+47*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[((20 + 67*x + 47*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] = ((20 + 67*x + 47*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.