Real-rooted atlas

A027466 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(8 + 19\,x + 11\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181119
264176121361
35122112290413316859
4409622528464644259214641130321
5327682252806195208518405856401610512476099
62621442162688743424013629440140553607730448177156147045881
7209715220185088832634881908121602623667202164525449920741619487171893871739
8167772161845493768881438722442395648419786752046176542723174637312124717894421435888116983563041
9134217728166094438491351941122930874777660449292288831177768967619129548844898441984154338394322357947691322687697779
1010737418241476395008091351941120334957117440805990563840132988443033615238259097601197291786240617353577280188635815280259374246016131066257801
1185899345921299227607048932189798403684528291840101324528025601950497164492826819336011776263404192972801810903826688082999758723202282493364888285311670611116490258898219
126871947673611338713661448574902206464393016351129601215894336307202674967539875844291093761884165057360505077764346169184051202655992279142401095596815146242738992037865631384283767212213314919066161
13549755813888982688517324881071802679296408737005174784140503345528832034774578018385926375339303370752876609154213478490400319028264966905579925770240379806895917363214242758596901123263965511789843452271214393142052983462257059
14439804651110484662395338752756670158340096416168587087052815736374699229184432750304228802568925475024719052814025746467415654416874726218609459215468499033725337610634593085686169653172965428430848182782068660231043866543760120272379749833583241799006685782884121
Technical details
Node id
n-61e603a5421ea593 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e348a50f4e54dede

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:         8,         11;
  2:        64,        176,        121;
  3:       512,       2112,       2904,        1331;
  4:      4096,      22528,      46464,       42592,       14641;
  5:     32768,     225280,     619520,      851840,      585640,      161051;
  6:    262144,    2162688,    7434240,    13629440,    14055360,     7730448,     1771561;
  7:   2097152,   20185088,   83263488,   190812160,   262366720,   216452544,    99207416,    19487171;
  8:  16777216,  184549376,  888143872,  2442395648,  4197867520,  4617654272,  3174637312,  1247178944,   214358881;
  9: 134217728, 1660944384, 9135194112, 29308747776, 60449292288, 83117776896, 76191295488, 44898441984, 15433839432, 2357947691;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (8+19*x+11*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[((8 + 19*x + 11*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] = ((8 + 19*x + 11*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.