Real-rooted atlas

A038220 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(5 + 16\,x + 11\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
151116
225110121256
3125825181513314096
4625550018150266201464165536
53125343751512503327503660251610511048576
6156252062501134375332750054903754831530177156116777216
778125120312579406252911562564054375845517756200463519487171268435456
8390625687500052937500232925000640543750112735700012400927007794868402143588814294967296
9195312538671875340312500174693750057648937501268276625018601390500175384539009646149645235794769168719476736
10976562521484375021269531251247812500048040781250126827662500232517381250292307565000241153741125117897384550259374246011099511627776
11488281251181640625129980468758578710937537746328125011625869062502557691193750401922901875044211519206253242178075125142655835305528531167061117592186044416
12244140625644531250077988281250571914062500283097460937599650306250002557691193750048230748225000663172788093756484356150250042796750591650171187002366603138428376721281474976710656
13122070312534912109375460839843750371744140625020445927734375809658738281252374998965625005224997724375008621246245218751053707874415625927262929485750556357757691450203997844486865345227121439314503599627370496
14610351562518798828125026882324218752365644531250014312149414062562973457421875020781240949218755224997724375000100581206194218751475191024181875016227101266000625129816810128005007139924557040275241658985007517037974983358324172057594037927936
Technical details
Node id
n-44fec6014b2adb13 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2235b7dd30e0ad6a

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:       5,       11;
  2:      25,      110,       121;
  3:     125,      825,      1815,       1331;
  4:     625,     5500,     18150,      26620,      14641;
  5:    3125,    34375,    151250,     332750,     366025,      161051;
  6:   15625,   206250,   1134375,    3327500,    5490375,     4831530,     1771561;
  7:   78125,  1203125,   7940625,   29115625,   64054375,    84551775,    62004635,    19487171;
  8:  390625,  6875000,  52937500,  232925000,  640543750,  1127357000,  1240092700,   779486840,  214358881;
  9: 1953125, 38671875, 340312500, 1746937500, 5764893750, 12682766250, 18601390500, 17538453900, 9646149645, 2357947691;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (5+16*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[((5 + 16*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] = ((5 + 16*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.