Real-rooted atlas

A038305 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(13 + 32\,x + 19\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 169 + 494\,x + 361\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01694943611024
12197963314079685932768
2285611669723660543566681303211048576
337129327132957931170115917108470865247609933554432
4482680942327402154657815301384460330363735193135722470458811073741824
56274851764196559728147722336856496465100210332958787675351428117517189387173934359738368
681573072195377745844878938537214261512647226054686567030463941216822262110889292962660856169835630411099511627776
71060449937313948995329181547972693227809949662046096796656678891070280591486822232467885438315660076198707687579732268769777935184372088832
8137858491849201485488087013251545562645516470493723601320972608946902316782729537642821722555206102356603452699601291599969268054194940071127061310662578011125899906842624
917921603940372881242479644121055233505091592319100753093526985583297058105521665505398042807012650788944684248573434010706156626520177705299938215085580587674247486554311649025889821936028797018963968
102329808512248140861256984043632846164267942741600197746386954052621887429263295123054259834584936209823289205125596262855549114113384240108434286930495155967871844501860683859130395123541817248038812216422133149190661611152921504606846976
1130287510659225357546270252528075046365237529384627043341913939522698812210839394409525995212390056067735065733696523746532740376463338086031681156650788982507316589642585430203585385240643455919594215355745927963228583740502213221812094205298346225705936893488147419103232
1239373763856992898056477835353929876536539435862333144744438439427210121798382237276978252952568096166557825854115245441595915233603192497880467902368216246175174059913605507239862990109659410494175284492772443415361931581919629769200523403857014031849001976536429901307847387990066857828841211180591620717411303424
135118589301409075711221522699242973651148048091537934996572709712464069216445318804123880918872105102507787524787760415324969845679114967280654692135836405120225265685773699166902186769577955471785639308410556836095218125293199079454146185819512485253522125070591207018512357462301915377515119551558063037276624035951518112702987479829937778931862957161709568
1466541660918317984115560511476283590128170567145028493199480116335539942510746312055259381472692604498201938329073195986742398451937279012559131943752108440472663585000761680178301161783394568560070231639116163042516419920236984634228343497568072188924813300917193865456115050367074276496264220517386266132841048288801620385558767688997388031576744222139580461922884414135676211676811208925819614629174706176
Technical details
Node id
n-79edb6541f685cb4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-de6ccc8c77edf67f

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           169,            494,             361;
  1:          2197,           9633,           14079,            6859;
  2:         28561,         166972,          366054,          356668,           130321;
  3:        371293,        2713295,         7931170,        11591710,          8470865,          2476099;
  4:       4826809,       42327402,       154657815,       301384460,        330363735,        193135722,         47045881;
  5:      62748517,      641965597,      2814772233,      6856496465,      10021033295,       8787675351,       4281175171,        893871739;
  6:     815730721,     9537774584,     48789385372,    142615126472,     260546865670,     304639412168,     222621108892,      92962660856,      16983563041;
  7:   10604499373,   139489953291,    815479726932,   2780994966204,    6096796656678,    8910702805914,    8682223246788,    5438315660076,    1987076875797,     322687697779;
  8:  137858491849,  2014854880870,  13251545562645,  51647049372360,  132097260894690,  231678272953764,  282172255520610,  235660345269960,  129159996926805,   41949400711270,   6131066257801;
  9: 1792160394037, 28812424796441, 210552335050915, 923191007530935, 2698558329705810, 5521665505398042, 8070126507889446, 8424857343401070, 6156626520177705, 2999382150855805, 876742474865543, 116490258898219;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (13+32*x+19*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 494*x + 361*x^2.
%t Mathematica
P[0, x_] := 169 + 494*x + 361*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((13 + 32*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] = 169 + 494*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((13 + 32*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

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.