Real-rooted atlas

A038327 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(13 + 28\,x + 15\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1131528
2169390225784
3219776058775337521952
42856113182022815017550050625614656
5371293214207549432505703750329062575937517210368
6482680933416370963933751482975001283343755923125011390625481890304
7627485175068149451754359425337376812538928093752695021875103654687517085937513492928512
8815730721752982204030408896700701743770001012130437509342742500053900437500177693750002562890625377801998336
910604499373110123647335508262987700136840035150023683852237502732752181250210211706250010395084375002998582031253844335937510578455953408
10137858491849159067490595082592735501252541314938500051315013181250710515567125006831880453125045045365625000194907832031254997636718750576650390625296196766695424
1117921603940372274665115508513123067974087545426004525687510482924121312501693395434981250195391780959375016103718210937509290606660156253573310253906258246100585937586497558593758293509467471872
12232980851224813225888709266602047198603957650787384078445250020441702036559375377385268367250005080186304943750050243600818125000362333659746093751858121332031250064319584570312501349361914062500129746337890625232218265089212416
133028751065922534543126598883795314524149153493501330679092572472503838497382420593757972263794258156251226502122193562500141519475637718750012246877699417968757850562627832031253623336597460937501140210817382812502192713110351562519461950683593756502111422497947648
14393737638569928963603772384373130477028292882798475220166904407445450069860652360054806251612168900616649375027902923279903546875367950636658068750003714886235490117187528576047965308593750164861815184472656256917278958789062500199536893041992187535420750244140625029192926025390625182059119829942534144
Technical details
Node id
n-b17ced2e7aeac1db · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-74360a37e7e7e405

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:          13,           15;
  2:         169,          390,          225;
  3:        2197,         7605,         8775,          3375;
  4:       28561,       131820,       228150,        175500,         50625;
  5:      371293,      2142075,      4943250,       5703750,       3290625,        759375;
  6:     4826809,     33416370,     96393375,     148297500,     128334375,      59231250,      11390625;
  7:    62748517,    506814945,   1754359425,    3373768125,    3892809375,    2695021875,    1036546875,     170859375;
  8:   815730721,   7529822040,  30408896700,   70174377000,  101213043750,   93427425000,   53900437500,   17769375000,   2562890625;
  9: 10604499373, 110123647335, 508262987700, 1368400351500, 2368385223750, 2732752181250, 2102117062500, 1039508437500, 299858203125, 38443359375;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (13+28*x+15*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[((13 + 28*x + 15*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] = ((13 + 28*x + 15*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

Candidate equalities ?

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.