Real-rooted atlas

A038335 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(12 + 33\,x + 21\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1122133
21445044411089
31728907215876926135937
4207361451523810244445281944811185921
5248832217728076204801333584011668860408410139135393
6298598431352832137168640320060160420078960294055272857661211291467969
7358318084389396482304433152672126336011762210880123503214247204354164180108854142618442977
8429981696601974374436870930432129048256512282293061120395210285568345808999872172904499936378228593611406408618241
95159780352812665405445688657838082322868617216609753012019210670677710336124491239953929336842996544408486881098879428004658146411484401953
1061917364224108355387392085329867571203982060486656012195060240384025609626504806437347371986176037347371986176024509212865928095313605589720166798809782011531578985264449
117430083706881430291113574412515047243776065703998029824022996399310438405634117831057408985970620435046412324632755438080107840536610083206290697968921520220174428912253235027750054222150542106513726817
128916100448256187238109413376180216680310374410512639684771840413935187587891201159018525246095362366329489044111363549494233566167043882259317962995203019535025082329601585255888168223045043996007807982473558275113866411667889514952984961
13106993205379072243409542237388825558001934925824163997179082440704717487658485678080226008612422988595252735342898697338889228685007272034304121126490720445450241177618659782108544082433306184747598083934316886090226272114750909177631599615447237773911946155040353993448503713
1412839184645488643145600238144716835781202708896153625046841896227307521205379266255939174442188274318957871104110744220087264411648221488440174528823296339154174017247260672395679869686788470784346219885975939911936220321745621052671232963907637092105436642595135946017206944832439199325215086811816331681783800622529
Technical details
Node id
n-fef6f335ad51810f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d36c4f1a2e9bd6b7

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:         12,          21;
  2:        144,         504,          441;
  3:       1728,        9072,        15876,          9261;
  4:      20736,      145152,       381024,        444528,        194481;
  5:     248832,     2177280,      7620480,      13335840,      11668860,        4084101;
  6:    2985984,    31352832,    137168640,     320060160,     420078960,      294055272,       85766121;
  7:   35831808,   438939648,   2304433152,    6721263360,   11762210880,    12350321424,     7204354164,    1801088541;
  8:  429981696,  6019743744,  36870930432,  129048256512,  282293061120,   395210285568,   345808999872,  172904499936,   37822859361;
  9: 5159780352, 81266540544, 568865783808, 2322868617216, 6097530120192, 10670677710336, 12449123995392, 9336842996544, 4084868810988, 794280046581;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (12+33*x+21*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[((12 + 33*x + 21*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] = ((12 + 33*x + 21*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.