Real-rooted atlas

A038277 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(29 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1291140
28416381211600
3243892775310527133164000
47072811073116610566154396146412560000
5205111493890045529510690111937102122945161051102400000
6594823321135373583412837150156492351801846962152802287417715614096000000
71724987630945801395717521188296093294868538512497777215284432171135962688319487171163840000000
85002464129611517989115192201526141154815288190018647248710784702199608789844171671842845210236722143588816553600000000
91450714597586949524394883139751404612020046650362658108437838270296134143524473537063629354503236589993589196559476679412357947691262144000000000
1042070723330020115957860573455902723841718572645275515024407348018288497309798108324419465149482631282014846105703271362228081124118514456838048303902593742460110485760000000000
1112200509765705829509055752293243219654505646940819510986161598243001583343294883222770442581634897114021678757925471811845483589113768308626197935369851090668704472058274038447719285311670611419430400000000000
12353814783205469041161046728907316942833597679651354051864247982484653960580362543332742019049522002629849170811289736795967736508443165657802318273687504792203771769512651756971875780143968268990310699288461372628313842837672116777216000000000000
13102606287129586021895059551399838207286311514841116873161410216014893967145431386615186537382637909073510368739316421744815752439601140523766884198908831912331529565658613321643914203119242809459929226518092012469782365418715874968740378118318749802381734522712143931671088640000000000000
142975582326757994634811580136821795624737106389585457787541961045159109517733282228572366165734177350991083841467745351385247599463826613097578815811693631153671225091522871448382899501431238194403968251612814625319181836339265682910088125328817457695311562401870620988348511401622113043598637974983358324126843545600000000000000
Technical details
Node id
n-a9d81a60d8ba037e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-194e35874175c670

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:             29,             11;
  2:            841,            638,            121;
  3:          24389,          27753,          10527,           1331;
  4:         707281,        1073116,         610566,         154396,          14641;
  5:       20511149,       38900455,       29510690,       11193710,        2122945,         161051;
  6:      594823321,     1353735834,     1283715015,      649235180,      184696215,       28022874,       1771561;
  7:    17249876309,    45801395717,    52118829609,    32948685385,    12497777215,     2844321711,     359626883,     19487171;
  8:   500246412961,  1517989115192,  2015261411548,  1528819001864,   724871078470,   219960878984,   41716718428,   4521023672,   214358881;
  9: 14507145975869, 49524394883139, 75140461202004, 66503626581084, 37838270296134, 14352447353706, 3629354503236, 589993589196, 55947667941, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (29+11*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[(29 + 11*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] = (29 + 11*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.