Real-rooted atlas

A038322 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 16\,x + 16\,n\right)\,P_{n}(x) = \left(361 + 608\,x + 256\,x^{2} + 304\,n + 256\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1351651
296911202562345
32418746512268804096101675
456929715479681488384573440655364244625
512901779455437606191872039690240114688001048576172571875
6284751385123857078421861004801981399040952565760220200960167772166880365625
7616301041131892155120693599639048161441792055479173120213374730244110417920268435456270225046875
8131399145633788865332608204109792768029593584599042611661373440142026683187245519942451275161927680429496729610487305390625
9276832077568318921476971152567983039477769797270052864010653690455654475215847555072340864039649289364102447104135291469824068719476736403085694921875
10577610979024414429313241092801513718157692160302924287721472039189080211456003409180945809408200575593480192077911780491264018728204894208024051816857600109951162777615369049541015625
1111955579202711951016595323082961638977956521616640888047985846067201332866865974476801379455623443251201000026410770759685043043493216256017140591708078080366240451264512042331197669376017592186044416582053491591796875
122457944462752420922954712069206944097593151015964313624945892173834649604262630332061122560511820876534199091244142579950184038402742929583828369408121033043837190144036566595643899904070318166642786304738871813865472028147497671065621916658723681640625
13502422486628018547511252448252503547223872900551975221760676645847044019240961297186393039401779201773254218137426984961774312371985223516161311665232805468569607131616917953760460827972081242372833280760585189393118003213296526056090501121280711144033484804503599627370496821150583457763671875
141021887498132846533711254263700467615452857260274204280397286417825099078808165580803789216743446507749376581139504081651997081666201490810463940771845677799590352715251712367266265185531199488017749802106907137146886265746198291514654721548828022036894842882482018197136893542422067638174115430407205759403792793630632633905533447265625
Technical details
Node id
n-b254c8c9f7d56cd1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3f787041db7ec6d3

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:            35,             16;
  2:           969,           1120,            256;
  3:         24187,          46512,          26880,           4096;
  4:        569297,        1547968,        1488384,         573440,           65536;
  5:      12901779,       45543760,       61918720,       39690240,        11468800,        1048576;
  6:     284751385,     1238570784,     2186100480,     1981399040,       952565760,      220200960,       16777216;
  7:    6163010411,    31892155120,    69359963904,    81614417920,     55479173120,    21337473024,     4110417920,     268435456;
  8:  131399145633,   788865332608,  2041097927680,  2959358459904,   2611661373440,  1420266831872,   455199424512,   75161927680,    4294967296;
  9: 2768320775683, 18921476971152, 56798303947776, 97972700528640, 106536904556544, 75215847555072, 34086403964928, 9364102447104, 1352914698240, 68719476736;
%F Formula
The row polynomials satisfy (16*n+3+16*x)*P(n,x) = (304*n+361+(256*n+608)*x+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[((304*n + 361 + (256*n + 608)*x + 256*x^2)*P[n - 1, x])/(16*n + 3 + 16*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] = ((304*n + 361 + (256*n + 608)*x + 256*x^2)*v[i - 1])/(16*n + 3 + 16*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.