Real-rooted atlas

A027467 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(16 + 33\,x + 17\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1161733
22565442891089
340961305613872491335937
465536278528443904314432835211185921
51048576557056011837440125772806681680141985739135393
616777216106954752284098560402472960320720640136306272241375691291467969
72684354561996488704636380774411269242880119735705607633151232270340772841033867342618442977
84294967296365072220161357612318722884926177283831542579203256811192321730180945925252335014469757574411406408618241
96871947673665712999628827928024842246923822825472110348426280961172452029235283048685404163781681210368100450907150411858787649746411484401953
1010995116277761168231104512055856049684480158258807439360294262470082560375184649355264332194741616640201689664552960803607257203201897406023952020159939004491531578985264449
111759218604441620560867439411210922960827187203481693763665920739859924779008011005416381087744116932549049057288874345240330240471449590892544016697173010777603548149264790243427189630763350542106513726817
12281474976710656358880595306086420972084788199424742761336248729601775663819469619203018628493098352643741841569569832963407748572286812162262958036284211201068619072689766403406223294198630465802040910655365826222372297611667889514952984961
134503599627370496622059698530549763965630578132254721544943579397357568410375638277423104078484340820557168641111861494957893222411813528383927615488941390543094231859255568191779867852802361648150644383744684341225470815744121185425343790288990457803290593755040353993448503713
1472057594037927936107185671131417804874025104125135421443146066925318255411291924142974142775296195338803820053397504311321218588210102272378032908285683695616351452469421846560768248945499173807980544132252296436085489664510974781684875755521357276763850451225622186254793709298881683778265594009291816331681783800622529
Technical details
Node id
n-904827c80385bef8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0b8e8228bfdf78a2

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,           17;
  2:         256,          544,           289;
  3:        4096,        13056,         13872,          4913;
  4:       65536,       278528,        443904,        314432,          83521;
  5:     1048576,      5570560,      11837440,      12577280,        6681680,        1419857;
  6:    16777216,    106954752,     284098560,     402472960,      320720640,      136306272,      24137569;
  7:   268435456,   1996488704,    6363807744,   11269242880,    11973570560,     7633151232,    2703407728,     410338673;
  8:  4294967296,  36507222016,  135761231872,  288492617728,   383154257920,   325681119232,  173018094592,   52523350144,    6975757441;
  9: 68719476736, 657129996288, 2792802484224, 6923822825472, 11034842628096, 11724520292352, 8304868540416, 3781681210368, 1004509071504, 118587876497;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (16+33*x+17*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 + 33*x + 17*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 + 33*x + 17*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

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.