Real-rooted atlas

A013622 → x ↦ x/(x+1) → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 14\,x + 11\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131114
2966121196
327297108913312744
48111886534159721464138416
5243445532670119790219615161051537824
6729160381470157187401976535289891817715617529536
7218756133617463377338513835745304386393720278119487171105413504
86561192456246985218112248830144702435091124464333724676921042143588811475789056
9196836495399526572815051164482781381643686506401790034863138434045787689787235794769120661046784
1059049216513035724645349307640224139069098621190363013425261063138434040868153468057073843073025937424601289254654976
1117714771449291309903651440894015105665561105424165469819888606722652089208083095496881485511671841070458559350118332853116706114049565169664
1253144123383404471565314576357606047549502495278957081304119331640335637504229819768594719333695140062092845401540683021299410271220141996313842837672156693912375296
1315943237599606316719133862247794663420604784414513599157713576648477104412243777493828446703881080282113656054052426520028879276892220028879276892212239870669211934522712143931793714773254144
1447829692455257425851696851858248871488654009454096346273599666349045047981631462664962970644692716756197471147108540403826210303232407368128040430987649082570372840534499144995391004510237974983358324111112006825558016
Technical details
Node id
n-255625afc7c906eb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4f0d71d8efeb5cf0

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:     3,     11;
  2:     9,     66,     121;
  3:    27,    297,    1089,     1331;
  4:    81,   1188,    6534,    15972,     14641;
  5:   243,   4455,   32670,   119790,    219615,     161051;
  6:   729,  16038,  147015,   718740,   1976535,    2898918,    1771561;
  7:  2187,  56133,  617463,  3773385,  13835745,   30438639,   37202781,   19487171;
  8:  6561, 192456, 2469852, 18112248,  83014470,  243509112,  446433372,  467692104,  214358881;
  9: 19683, 649539, 9526572, 81505116, 448278138, 1643686506, 4017900348, 6313843404, 5787689787, 2357947691;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3+14*x+11*x^2)*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[Together[((3 + 14*x + 11*x^2)*P[n - 1, x])/(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] = ((3 + 14*x + 11*x^2)*v[i - 1])/(1 + x)); 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

Candidate equalities ?

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.