Real-rooted atlas

A038324 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(31 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1312051
296112404002601
32979157660372008000132651
4923521238328023064009920001600006765201
5286291519235210011916400076880000248000003200000345025251
688750368134354981205541126000476656000023064000005952000006400000017596287801
72751261411112425051534024048486840025858588000016682960000064579200000138880000001280000000897410677851
8852891037441440201825776099400412272001282585964800010343435200000533854720000017221120000003174400000002560000000045767944570401
9264396221606711535203867393803961816431984005964024736320005771636841600003723636672000001601564160000004428288000000071424000000005120000000002334165173090451
10819628286980801528792443213420015352038673938000264121095465600002982012368160000023086547366400000124121222400000004575897600000000110707200000000015872000000000010240000000000119042423827613001
11254084768964048311803182231357762205816716875347620001125816169422120000145266602506080000013120854419904000008465067367680000003900952704000000001258371840000000002706176000000000034918400000000002048000000000006071163615208263051
127876627837885497616098034455137159440216381867762931464004653373500278096000067548970165327200000697279692029184000005248341767961600000029023088117760000000117028581120000000003355658240000000000649482240000000000761856000000000004096000000000000309629344375621415601
132441754629744504259120479232378502293786079274447916783072720018753095206120726880003024692775180762400000351254644859701440000030215453321264640000001949384085242880000000943250363827200000000338082567680000000000872471142400000000001535139840000000000016506880000000000008192000000000000015791096563156692195651
1475694393522079632032168369129632846119254802867092532990321130040073989484722330867872000131271666442845088160000169382795410122694400000163918834267860672000000120861813285058560000000682284429835008000000002934556687462400000000094663118950400000000002220835635200000000000358199296000000000000355532800000000000001638400000000000000805345924720991301978201
Technical details
Node id
n-0996adca1233ccf5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-74659e42bd6f64b0

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:             31,              20;
  2:            961,            1240,             400;
  3:          29791,           57660,           37200,            8000;
  4:         923521,         2383280,         2306400,          992000,          160000;
  5:       28629151,        92352100,       119164000,        76880000,        24800000,         3200000;
  6:      887503681,      3435498120,      5541126000,      4766560000,      2306400000,       595200000,        64000000;
  7:    27512614111,    124250515340,    240484868400,    258585880000,    166829600000,     64579200000,     13888000000,     1280000000;
  8:   852891037441,   4402018257760,   9940041227200,  12825859648000,  10343435200000,   5338547200000,   1722112000000,   317440000000,   25600000000;
  9: 26439622160671, 153520386739380, 396181643198400, 596402473632000, 577163684160000, 372363667200000, 160156416000000, 44282880000000, 7142400000000, 512000000000;
%F Formula
The row polynomials satisfy P(n,x) = (31+20*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[(31 + 20*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] = (31 + 20*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.