Real-rooted atlas

A147716 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(17 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
117219
2289684361
34913173420486859
48352139304693654416130321
51419857835210196520231201360322476099
6241375691703828450112607860806936032646447045881
74103386733379259661192679882338588027512801942087616128893871739
86975757441656541876827034077286360959369354352088040965178881740825616983563041
91185878764971255636339385908876891216220446368286243171233675667226412288133171239168512322687697779
10201599390044923717575299401255636339380393925126080811022318401144972684811225222407546368033292808704010246131066257801
113427189630763344351865809878260893328293409207999822120216658819344035684982009641982331776352792704020752512081382401914882048116490258898219
1258262223722976182252551138319253222238971853620871466263472055247998932720103996233285121427399280384143939423232105837811205534003201953177641779240962213314919066161
13990457803290593715148178167973786106928316479814964612594044227312135664530712568028728959445014445065034423552530176875571246780312550430575367680143884083246166016905216819242052983462257059
14168377826559400929277328184921366236212074494351633004997997620478272963228815830959118475972137199038081340684774100672180260137694208185561906449921455387501568856110295043662503936107720704194969616384799006685782884121
Technical details
Node id
n-2ebbd73e6940b9ed · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0d25933118d4a2ef

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:           17,            2;
  2:          289,           68,           4;
  3:         4913,         1734,         204,           8;
  4:        83521,        39304,        6936,         544,         16;
  5:      1419857,       835210,      196520,       23120,       1360,        32;
  6:     24137569,     17038284,     5011260,      786080,      69360,      3264,       64;
  7:    410338673,    337925966,   119267988,    23385880,    2751280,    194208,     7616,     128;
  8:   6975757441,   6565418768,  2703407728,   636095936,   93543520,   8804096,   517888,   17408,   256;
  9: 118587876497, 125563633938, 59088768912, 16220446368, 2862431712, 336756672, 26412288, 1331712, 39168, 512;
%F Formula
The row polynomials satisfy P(n,x) = (17+2*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[(17 + 2*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] = (17 + 2*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

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.