Real-rooted atlas

A038271 → x ↦ x/(x+1) → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(7 + 26\,x + 19\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
171926
249266361676
334327937581685917576
4240126068106134192052130321456976
516807228095123823033609104561235247609911881376
6117649191599813001415470527409578593510399615847045881308915776
7823543156473171274138675763960651564503605254790587123052481698938717398031810176
85764801125178536118919609264556359282190305047047560909592645469487325005681738416983563041208827064576
94035360798578097110702764828677841772442759784359227490843260741355485923372157678974759610699644715833226876977795429503678976
10282475249766718533093649192245677841772440321974841909010487180565036237210036590103679176077724037448756505405225881388445306131066257801141167095653376
111977326743590373270418012208669856524227059735354172326099901345854839179623653034563487547082413949618709611847503053958696433455144054720921018506771164902588982193670344486987776
121384128720145083049740467302552826746089278589086037188094240489516150258070155445114248388882556118984554353594162018487975641329524350013674403340198278682777284349785181747450396221331491906616195428956661682176
139688901040734187979386475567756642939455412435160682637601295287606051837091855480181366485229055473228180459907436284476367364811566721969553962811092675985601445337757762498445225769508993018201411657635020651420529834622570592481152873203736576
1467822307284925772476768262454700125840051493674422340626836849269381853929200038890930064186814444055929547043252643870410798266460002919222564588271085767109741644930614735410775065181201145440418039604385889869171224116011899412119237930119178279900668578288412164509974703297150976
Technical details
Node id
n-2f231570893cd592 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8f7225f9e9fdccc1

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:        7,        19;
  2:       49,       266,         361;
  3:      343,      2793,        7581,        6859;
  4:     2401,     26068,      106134,      192052,       130321;
  5:    16807,    228095,     1238230,     3360910,      4561235,      2476099;
  6:   117649,   1915998,    13001415,    47052740,     95785935,    103996158,      47045881;
  7:   823543,  15647317,   127413867,   576396065,   1564503605,   2547905871,    2305248169,     893871739;
  8:  5764801, 125178536,  1189196092,  6455635928,  21903050470,  47560909592,   64546948732,   50056817384,   16983563041;
  9: 40353607, 985780971, 10702764828, 67784177244, 275978435922, 749084326074, 1355485923372, 1576789747596, 1069964471583, 322687697779;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (7+26*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[((7 + 26*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] = ((7 + 26*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

Same fingerprint ?

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.