Real-rooted atlas

A013610 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(16 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1161228
2256384144784
3409692166912172821952
46553619660822118411059220736614656
51048576393216058982404423680165888024883217210368
6167772167549747214155776014155776079626240238878722985984481890304
7268435456140928614431708938243963617280297271296013377208323344302083583180813492928512
8429496729625769803776676457349121014686023689512681472057076088832214035333124586471424429981696377801998336
9687194767364638564679681391569403904243524645683227396522639362054739197952102736959897633022594252861917364224515978035210578455953408
1010995116277768246337208320278313880780805566277615616073057393704960657516543344644109478395904017612050268160495338913792082556485632061917364224296196766695424
1117592186044416145135534866432544258255749120122458107543552018368716131532801928715193810944144653639535820877493021179904029059882942464072649707356160108974561034247430083706888293509467471872
12281474976710656253327479039590410449758510383104261243962759577604408491871567872052901902458814464462891646514626562975732013308313613948743812382720464958127079424010461557859287041426576071720968916100448256232218265089212416
134503599627370496439100963668623361975954336508805125433874425399214081018851454762352640137544946392917606413754494639291760641031587097946882048580267742595121152241778226081300480725334678243901441483639114589798418545488932372481069932053790726502111422497947648
147205759403792793675660473739824332836884480948164362241106534428444930867222822272586676699136342334088800150487043851258499001692979233010787134300225536216633290568845230081083166452844226150440618741981658480641107783872227049472207709476042571776239664780049121281283918464548864182059119829942534144
Technical details
Node id
n-2f6af0c5a014525d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c91d5d506c107735

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:          16,           12;
  2:         256,          384,           144;
  3:        4096,         9216,          6912,          1728;
  4:       65536,       196608,        221184,        110592,         20736;
  5:     1048576,      3932160,       5898240,       4423680,       1658880,        248832;
  6:    16777216,     75497472,     141557760,     141557760,      79626240,      23887872,       2985984;
  7:   268435456,   1409286144,    3170893824,    3963617280,    2972712960,    1337720832,     334430208,     35831808;
  8:  4294967296,  25769803776,   67645734912,  101468602368,   95126814720,   57076088832,   21403533312,   4586471424,   429981696;
  9: 68719476736, 463856467968, 1391569403904, 2435246456832, 2739652263936, 2054739197952, 1027369598976, 330225942528, 61917364224, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (16+12*x)*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[(16 + 12*x)*P[n - 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] = (16 + 12*x)*v[i - 1]); 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.