Real-rooted atlas

A038322 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(19 + 16\,n x\right)\,P_{n}(x) = \left(361 + 608\,x + 256\,x^{2} + 304\,n x + 256\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1193251
236112167682345
36859346564377616384101675
4130321877952166348812451843276804244625
52476099208513605267712059146240311296006291456172571875
6470458814754110081501297920224755712017743872007172259841174405126880365625
78938717391053827734439934524672747312742407866449920047695527936156195880962147483648270225046875
816983563041228831165184101167462502422718307368962989250969600241657341542411870886952963264175144963865470566410487305390625
93226876977794891266155808247137658398726474717600153610223238316032010330851350937667664055631872279086974894086609954668544687194767360403085694921875
106131066257801103260063289280586951938696960175742334861312032373588000768003925723513356288321404264251392017675508409958406279456935116801305670057984001209462790553615369049541015625
1111649025889821921581353227459521363032835418496045912684982517760966582841737216001367460357152440321343469824570818569235453144203264043746883314647040136442521059328002527777232257024211106232532992582053491591796875
12221331491906616144732259416916096310771486475417088116312135289044992027547610989510656004454013734725091328510518533336911052842113666337566883842493572348934881280103696316005089280028816660447730073648132221017522176365917469723852821916658723681640625
1342052983462257059920739006331522976697823246903891097628729097416394113024756028879378792448001375176740596371947521801401110488814714881733679264229836718081231824740373831352326403247513314263040023725717101964427264594432929566398873690381615021791641663050394783186944821150583457763671875
1479900668578288412118839736591091162432154684153063695859968694721810250984914944201103681914758791168040644112555403882004485989658692375308926976658798120407337952870454610896823239856619523406527677083187937280157776018728063441305652706386421554033459212020754797898288332816771405012327727104108086391056891904030632633905533447265625
Technical details
Node id
n-3c3a73ee67796c77 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-79c548f63bca45cd

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:           19,            32;
  2:          361,          1216,            768;
  3:         6859,         34656,          43776,          16384;
  4:       130321,        877952,        1663488,        1245184,          327680;
  5:      2476099,      20851360,       52677120,       59146240,        31129600,         6291456;
  6:     47045881,     475411008,     1501297920,     2247557120,      1774387200,       717225984,      117440512;
  7:    893871739,   10538277344,    39934524672,    74731274240,     78664499200,     47695527936,    15619588096,     2147483648;
  8:  16983563041,  228831165184,  1011674625024,  2271830736896,   2989250969600,   2416573415424,  1187088695296,   326417514496,   38654705664;
  9: 322687697779, 4891266155808, 24713765839872, 64747176001536, 102232383160320, 103308513509376, 67664055631872, 27908697489408, 6609954668544, 687194767360;
%F Formula
The row polynomials satisfy (19+16*n*x)*P(n,x) = (361+(304*n+608)*x+(256*n+256)*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[((361 + (304*n + 608)*x + (256*n + 256)*x^2)*P[n - 1, x])/(19 + 16*n*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] = ((361 + (304*n + 608)*x + (256*n + 256)*x^2)*v[i - 1])/(19 + 16*n*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

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.