Real-rooted atlas

A225471 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(x + x^{2} + 4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1459
2327645153
3384132815285853825
461442777647024353369945126225
5122880684544152377616942249409562088455175225
6294912019501056536842247875577664938544285361805221125253586025
782575360631554048206868889637620042244102196736268223081697374072015141262514454403425
826424115202293471641687039328256188650868736255416434944221203878400119673319968369786477604996616625939536222625
9951268147209234190172163982000324608100118864199681617507380121617413747715328124927830196485759144158176154808816562018487481512568586144251625
103805072588800408369600921601971401926901765637374701076481057490295267328135972793446297612136749771143687425698701326082980484371100167086660739542075798674201255555477684381625
1116742319390720019680545105510401051183168251494433675757355794432718977591466065921074150924034007041145894560438661128728844825598131246529775388808064165303103953492003522511499279400341094033905625494437513909964625
128036313307545600102670353007902720601002591777718272213151610552136499251012045494710435848678914633546956800107636334178722078729804728862436542464651056318320587136030734138930283567369790657613375220001889755770921606001671360766137562547960438849266568625
134178882919923712005764782961710858240366936634818601943041426919748513300480003782329901651266109447216674020664070635521019691413307343519744108031847199392722534485819991523584206131250487737114738573235221380235592005405100861717215359681017200108848131842760443008858212060529081255035846079172989705625
1423401744351572787200346647478499373219840238343778380168980070410082289410140513763328293144900157729029488646197265495513101238272098237761062996439728128118620244990178505244672109637348156861007278080771905279526965985064964075094208692401360051215642898347585217870656412743081382127646120067004033904269738010050491808745015763125569050606946547836735625
Technical details
Node id
n-1e8cd0107cd0683d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9ac0ab5e73a51d73

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:           4,            5;
  2:          32,           76,            45;
  3:         384,         1328,          1528,            585;
  4:        6144,        27776,         47024,          35336,           9945;
  5:      122880,       684544,       1523776,        1694224,         940956,         208845;
  6:     2949120,     19501056,      53684224,       78755776,       64938544,       28536180,        5221125;
  7:    82575360,    631554048,    2068688896,     3762004224,     4102196736,     2682230816,      973740720,     151412625;
  8:  2642411520,  22934716416,   87039328256,   188650868736,   255416434944,   221203878400,   119673319968,   36978647760,    4996616625;
  9: 95126814720, 923419017216, 3982000324608, 10011886419968, 16175073801216, 17413747715328, 12492783019648, 5759144158176, 1548088165620, 184874815125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n+(8*n+1)*x+(4*n+1)*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[((4*n + (8*n + 1)*x + (4*n + 1)*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] = ((4*n + (8*n + 1)*x + (4*n + 1)*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

Same fingerprint ?

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.