Real-rooted atlas

A027467 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(16 + 35\,x + 19\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1161935
22566083611225
340961459217328685942875
4655363112965544964389761303211500625
510485766225920147865601755904010425680247609952521875
616777216119537664354877440561889280500432640237705504470458811838265625
726843545622313697287949254656157328998401868281856013311508224526913867289387173964339296875
8429496729640802189312169584099328402762235904597850193920567957684224337224875008114415582592169835630412251875390625
968719476736734439407616348858718617696662936616961721808558489620446476632064161867940003848237921946624244563307790432268769777978815638671875
10109951162777613056700579840697717437235202209438551244804591489489305606542872522260486474717600153604393558371532801956506462323205163003164464061310662578012758547353515625
11175921860444162297979302051841364425210593280486076481273856011544316430254080191924260652974082279100595254067219331656834744320114781712456294404543442784728320107906766137297611649025889821996549157373046875
122814749767106564011018418126848261969640433909761036963160050892802770635943260979205264208292195860487293121904813015047423356224541818885509522197902131202907803382226124801035904954918056962236612970845804822133149190661613379220508056640625
134503599627370496695243192475320324953607746386657282156883372905857024640324751331426304013686941559709237248216709908028729589762573430157841163878422919612343272865792151205775875758489607182274354098528256232607748967963699246036950316576148842052983462257059118272717781982421875
147205759403792793611979575008805519369246734459921760256439219886846283612161434327442982394920963406527677083187937286067877424804428513288234976505091724410888556655274821869895686774018759233980334084022073638295175823361736804525627462287365156138435456528665694198682955455812167990066857828841214139545122369384765625
Technical details
Node id
n-29ea66a26bb11d2c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4b6c87ac223a4827

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:          16,           19;
  2:         256,          608,           361;
  3:        4096,        14592,         17328,          6859;
  4:       65536,       311296,        554496,        438976,         130321;
  5:     1048576,      6225920,      14786560,      17559040,       10425680,        2476099;
  6:    16777216,    119537664,     354877440,     561889280,      500432640,      237705504,       47045881;
  7:   268435456,   2231369728,    7949254656,   15732899840,    18682818560,    13311508224,     5269138672,     893871739;
  8:  4294967296,  40802189312,  169584099328,  402762235904,   597850193920,   567957684224,   337224875008,  114415582592,   16983563041;
  9: 68719476736, 734439407616, 3488587186176, 9666293661696, 17218085584896, 20446476632064, 16186794000384, 8237921946624, 2445633077904, 322687697779;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (16+35*x+19*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[((16 + 35*x + 19*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] = ((16 + 35*x + 19*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.