Real-rooted atlas

A204420 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + x - 4\,n - 6\,n x - 2\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
29302463
3225102015007203465
4110256258013062011928040320363825
58930256061230162124202141748014001120362880062214075
6108056025851288130276178518047315545204521242880228723264047900160015740160975
718261468225163533890520621676135080130227761664016252329693601209408520320497227610880871782912005524796502225
8410883035062541177877741000179125264117800442214736163200678224046099600662172829718400402134257324800138949742131200209227898880002569030373534625
9118745197133062513157708754440250643676319187902001826123895712116003313244585887236003989045058950952003188416862606976001632095582173056004856530736977920064023737057280001528573072253101875
1042867016165035562552011646094585677502823664444937055720090382772764347663600189000837587589627600269907820891844335200266685560980248009600180078491295512409600795517080830664192002076607370828390400024329020081766400001132672646539548489375
1118904354128780683062524917592074536926765001485529825174227402570052904132524686517074000125106210393255646354800206347735978769759774400242305745644321455096000202623343834195553068800118278566243159024102400459107276397298873344001066683583883306188800011240007277776076800001022803399825212285905625
12100004033341249813400625142249265553387279637350092339038576861013170233003618641074052090279433240095384266451659194346537200178216580469846319708507200242083689706126957141252800240940520483954890786387200174417447339107069044147200895765434876408917044224003098547781587358754734080064826897680302046248960006204484017332394393600001105650475211054481063980625
136250252083828113337539062595406053138048287644384375066958101371508306407823375002861854422032153010252289500083136333513625583282901810000173385135987232426143066180000267143083371729456023812680000307942223611454328883306320000265622242901512597163743680000169356680450195152193959680000775906769018875718249625600002419224168533670979633152000046015286003029076557824000004032914611266056355840000001409704355894094463356575296875
144556433769110694623065976562574276203313011255375935740625056025153517066005917218699125002592495120030042340917063760500082242006561996126970742728110000189248832270993379620169013580000325826470584338487605517475800000426450052041777674509932693360000426442936125462155968148720640000324271435681735789405186828800000184597253881823574919031224320000762946959264374657411973734400002164384906373537228709199872000037727550969902936961122304000003048883446117138605015040000002093410968502730278084514315859375
Technical details
Node id
n-d75f4aef92e17cce · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f03d443003d0524d

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:                1,                 2;
  2:                9,                30,                24;
  3:              225,              1020,              1500,                720;
  4:            11025,             62580,            130620,             119280,              40320;
  5:           893025,           6061230,          16212420,           21417480,           14001120,            3628800;
  6:        108056025,         851288130,        2761785180,         4731554520,         4521242880,         2287232640,          479001600;
  7:      18261468225,      163533890520,      621676135080,      1302277616640,      1625232969360,      1209408520320,       497227610880,        87178291200;
  8:    4108830350625,    41177877741000,   179125264117800,    442214736163200,    678224046099600,    662172829718400,    402134257324800,    138949742131200,    20922789888000;
  9: 1187451971330625, 13157708754440250, 64367631918790200, 182612389571211600, 331324458588723600, 398904505895095200, 318841686260697600, 163209558217305600, 48565307369779200, 6402373705728000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n^2-4*n+1+(8*n^2-6*n+1)*x+(4*n^2-2*n)*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^2 - 4*n + 1 + (8*n^2 - 6*n + 1)*x + (4*n^2 - 2*n)*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^2 - 4*n + 1 + (8*n^2 - 6*n + 1)*x + (4*n^2 - 2*n)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.