Real-rooted atlas

A038291 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11910
211881100
31272437291000
41364862916656110000
514581072903280559049100000
6154121514580984153542945314411000000
716317012551522963512400293720087478296910000000
81722268408244592703306744148803483826375243046721100000000
91812916612368266867440174446410441721868843874204893874204891000000000
1019036458748013778101488034811160261057395628019371024453874204890348678440110000000000
11199445512028521651302728063824552574215783797707102708965213081268953835462841131381059609100000000000
1211085346160380324769546766808491051484378811144821308126895852325075802301277704663765727153082824295364811000000000000
131117631820849446911157599606391195275682075748045540112992727700564963599722033868624477226495023671583974253254186582832910000000000000
14112673712653566567561118216098159591732316415149608129269303163775615818978349027118540111422705697676257010878197713558612159660622876792454961100000000000000
Technical details
Node id
n-5622cca49392ecf7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e9bc54cd164ac143

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: 1,  9;
  2: 1, 18,   81;
  3: 1, 27,  243,   729;
  4: 1, 36,  486,  2916,   6561;
  5: 1, 45,  810,  7290,  32805,   59049;
  6: 1, 54, 1215, 14580,  98415,  354294,   531441;
  7: 1, 63, 1701, 25515, 229635, 1240029,  3720087,   4782969;
  8: 1, 72, 2268, 40824, 459270, 3306744, 14880348,  38263752,  43046721;
  9: 1, 81, 2916, 61236, 826686, 7440174, 44641044, 172186884, 387420489, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (1+9*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[(1 + 9*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] = (1 + 9*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.