Real-rooted atlas

A027467 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(19 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
119322
23611149484
3685932495132710648
41303218230819494205281234256
5247609919548156173109747076952435153632
6470458814456978217593335370386043861527702729113379904
78938717399879635014679827111231533451944526518421839695721872494357888
81698356304121452921736118555620123743861688738920070933372727368732332424656154875873536
9322687697779458556202107289614443436106700058108252710663943990168378420017724284222521121931196831207269217792
10613106625780196806309333706878343031605289614443436080025043581015162639836419950841890180007596010658344537397705904926559922791424
111164902588982192023251865074331597304104006057566177334765523893191583470528165287634683394519100294053968910742531333539080596512341241177147584318301411328
1222133149190661614193649320335884364185335713379419167649248072606809559601288951720309794009843168991725807642888609822964232428600952970125334014069014744038951653144112855002631049216
1342052983462257059863192818435802798177616174654973847344093642739322186884580168707855311456489005381111820136610639617655811043785220908197288693183405239374511583488802649881052261312659271594323282810057883082752
147990066857828841211766225305414796478181270491871518585911448662644516963324971129832487628811569830473417145943718019542303766767092081966383769269300797987239757158734723477030200537294422786633721745836829142408991847829696221821273427820544
Technical details
Node id
n-9b494e433b302504 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-40f5512fcb1be880

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,            3;
  2:          361,          114,            9;
  3:         6859,         3249,          513,           27;
  4:       130321,        82308,        19494,         2052,          81;
  5:      2476099,      1954815,       617310,        97470,        7695,        243;
  6:     47045881,     44569782,     17593335,      3703860,      438615,      27702,       729;
  7:    893871739,    987963501,    467982711,    123153345,    19445265,    1842183,     96957,     2187;
  8:  16983563041,  21452921736,  11855562012,   3743861688,   738920070,   93337272,   7368732,   332424,    6561;
  9: 322687697779, 458556202107, 289614443436, 106700058108, 25271066394, 3990168378, 420017724, 28422252, 1121931, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (19+3*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[(19 + 3*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] = (19 + 3*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.