Real-rooted atlas

A013622 → x ↦ x/(x+1) → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(11 + 32\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1113243
212170410241849
3133111616337923276879507
414641170368743424144179210485763418801
5161051234256013629440396492805767168033554432147008443
61771561309217922248857608722841601903165440221459251210737418246321363049
71948717139682966434632407041679147008048847912960852618117128267812044834359738368271818611107
82143588814988715776507941969922955298734081074654085120250101314355236378372997123023656976384109951162777611688200277601
92357947691617353577287183750717444876242911232212781508853766190007530291212004863089049614967102033100810885165114982435184372088832502592611936843
102593742460175454326112098776572364807662667431936039009943289856013618016566640643301337349488640548793741213696059868408132403203870280929771520112589990684262421611482313284249
11285311670611912997345955213279961395712011589784490803206743147340103680274630000760586247989236385762508816601010671714304024146924613402624023415199625117696013623388872795750436028797018963968929293739471222707
1231384283767211095596815146241752954904233984169983505865113601112619311117107205178737157199626241757632004867751936438266681733257625679684851224228659201030268783505178624089914366560451952644755801206503243776115292150460684697639959630797262576401
13345227121439311305586204715936227884137550417922430764133871124481767828460997181440925699266849433190435905910956584075264104453559146426400768227898674501293965312368321090103101358080428591813938154307584340039786264981929984164867775158779117568368934881474191032321718264124282290785243
143797498335832411546617504048108829245130985636966434030697874195742722722455829935659417615839743010534745702469118878591424344883222979783012213808168965849399312199878443008113442895751755218288641650078483661894084198417455375694935739072512126948186872259920527365681597174702541897728118059162071741130342473885357344138503765449
Technical details
Node id
n-5ac825d4e77da5ac · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-e63d418fb4410416

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:         11,          32;
  2:        121,         704,         1024;
  3:       1331,       11616,        33792,         32768;
  4:      14641,      170368,       743424,       1441792,        1048576;
  5:     161051,     2342560,     13629440,      39649280,       57671680,       33554432;
  6:    1771561,    30921792,    224885760,     872284160,     1903165440,     2214592512,      1073741824;
  7:   19487171,   396829664,   3463240704,   16791470080,    48847912960,    85261811712,     82678120448,     34359738368;
  8:  214358881,  4988715776,  50794196992,  295529873408,  1074654085120,  2501013143552,   3637837299712,   3023656976384,   1099511627776;
  9: 2357947691, 61735357728, 718375071744, 4876242911232, 21278150885376, 61900075302912, 120048630890496, 149671020331008, 108851651149824, 35184372088832;
%F Formula
The row polynomials satisfy P(n,x) = (11+32*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[(11 + 32*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] = (11 + 32*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.