Real-rooted atlas

A147716 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(15 + 34\,x + 19\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1151934
22255703611156
333751282516245685939304
4506252565004873504115401303211336336
5759375480937512183750154327509774075247609945435424
61139062586568750274134375462982500439833375222848910470458811544804416
717085937515149531255756821875121532906251539416812511699567775493981750589387173952523350144
8256289062525970625000115136437500291678975000461825043750467982711000296389050300107264608680169835630411785793904896
938443359375438254296875222048843750065627769375001246927618125015794416496250133375072635007240361085900229278101053532268769777960716992766464
105766503906257304238281250416341582031251406309343750003117319045312504738324948875005001565223812503620180542950001719585757901254840315466685061310662578012064377754059776
1186497558593751205199316406257632929003906252900513021484375734796632109375013030393609406250165051652385812501493324473966875094577216684568753993260260015125101162593253716511649025889821970188843638032384
1212974633789062519721443359375001373927220703125058010260429687500165329242224609375335067264241875000495154957157437500537596810628075000425597475080559375239595615600907500910463339283448502096824660167942022133149190661612386420683693101056
1319461950683593753204734545898437524355982548828125011312000783789062503582133581533203125816726456589570312513793602377957187500174718963454124375001659830152814181562511680286260544240625591801170534241525020444040436637434504315964092179013954205298346225705981138303245565435904
1429192926025390625517687888183593750426229694604492187521595637859960937500752248052121972656251905695065375664062503620820624213761718755241568903623731250005809405534849635468754905720229428581062503106956145304768006251431082830564620415004531762296787964647588311265270739823907990066857828841212758702310349224820736
Technical details
Node id
n-ef5f3a4b674fad08 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-74fe1f9c17c9cb0a

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:          15,           19;
  2:         225,          570,           361;
  3:        3375,        12825,         16245,          6859;
  4:       50625,       256500,        487350,        411540,         130321;
  5:      759375,      4809375,      12183750,      15432750,        9774075,        2476099;
  6:    11390625,     86568750,     274134375,     462982500,      439833375,      222848910,       47045881;
  7:   170859375,   1514953125,    5756821875,   12153290625,    15394168125,    11699567775,     4939817505,     893871739;
  8:  2562890625,  25970625000,  115136437500,  291678975000,   461825043750,   467982711000,   296389050300,  107264608680,   16983563041;
  9: 38443359375, 438254296875, 2220488437500, 6562776937500, 12469276181250, 15794416496250, 13337507263500, 7240361085900, 2292781010535, 322687697779;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (15+34*x+19*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[((15 + 34*x + 19*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] = ((15 + 34*x + 19*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.