Real-rooted atlas

A038305 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(13 + 38\,x + 25\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 169 + 650\,x + 625\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01696506251444
1219712675243751562554872
2285612197006337508125003906252085136
33712933570125137312502640625025390625976562579235168
44826809556939502677593756865625009902343757617187502441406253010936384
5627485178446915754873220625156192968753003710937534658203125222167968756103515625114415582592
68157307211254970340084469157500324881375000780964843750120148437500011552734375006347656250001525878906254347792138496
7106044993731835394122251411841632500633518681250018274577343750351434179687504505566406250037133789062500178527832031253814697265625165216101262848
8137858491849265112484325022942426528125117653469375000395949175781250913728867187500146430908203125016091308593750001160430908203125495910644531250953674316406256278211847988224
9179216039403737911085258475364529665946875210305576507812580886760195312502177720466796875041879239746093750575264282226562505531387329101562535457611083984375136375427246093752384185791015625238572050223552512
102329808512248153764811821110056866627887712503645296659468750015772918238085937548532056117187500010888602333984375001794824560546875000215724105834960937518437957763671875001063728332519531250371932983398437500596046447753906259065737908494995456
11302875106592253757187766480632587367819209303750616055135450218750296180353581835937510252396854755859375262881970634765625005055422512207031250072914747772216796875779003715515136718755992336273193359375031428337097167968750100731849670410156251490116119384765625344498040522809827328
12393737638569928910600628730728855013250785913411068751019291224108543750053904824351894140625207326247507285156250598056483194091796875131440985317382812500022117473490905761718752835573524475097656250272651300430297851562519066524505615234375009166598320007324218752712011337280273437503725290298461914062513090925539866773438464
13511858930140907571476516144637233375198761788701166031251656348239176383593759555855226017597656254042861826392060546875129578904692053222656253203874017111206054687561612961867523193359375921561395454406738281251063340071678161621093759294930696487426757812559582889080047607421875264421105384826660156257264316082000732421875931322574615478515625497455170514937390661632
1466541660918317984120474357205636302800295303228927446675000265015718268221375000016563482391763835937500764468418081407812500002695241217594707031250007404508839545898437500001601937008555603027343750273835386077880859375000036862455818176269531250003866691169738769531250000309831023216247558593750018333196640014648437500007554888725280761718750001937150955200195312500002328306436538696289062518903296479567620845142016
Technical details
Node id
n-467652f7158f8190 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-93ef29b2c9892602

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,            650,             625;
  1:          2197,          12675,           24375,            15625;
  2:         28561,         219700,          633750,           812500,           390625;
  3:        371293,        3570125,        13731250,         26406250,         25390625,           9765625;
  4:       4826809,       55693950,       267759375,        686562500,        990234375,         761718750,         244140625;
  5:      62748517,      844691575,      4873220625,      15619296875,      30037109375,       34658203125,       22216796875,        6103515625;
  6:     815730721,    12549703400,     84469157500,     324881375000,     780964843750,     1201484375000,     1155273437500,      634765625000,      152587890625;
  7:   10604499373,   183539412225,   1411841632500,    6335186812500,   18274577343750,    35143417968750,    45055664062500,    37133789062500,    17852783203125,     3814697265625;
  8:  137858491849,  2651124843250,  22942426528125,  117653469375000,  395949175781250,   913728867187500,  1464309082031250,  1609130859375000,  1160430908203125,   495910644531250,    95367431640625;
  9: 1792160394037, 37911085258475, 364529665946875, 2103055765078125, 8088676019531250, 21777204667968750, 41879239746093750, 57526428222656250, 55313873291015625, 35457611083984375, 13637542724609375, 2384185791015625;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (13+38*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 650*x + 625*x^2.
%t Mathematica
P[0, x_] := 169 + 650*x + 625*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((13 + 38*x + 25*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 + 650*x + 625*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((13 + 38*x + 25*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.