Real-rooted atlas

A059364 → basis A013610 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 6\,n x - 9\,n x^{2} + 4\,n^{2} + 18\,n^{2} x + 18\,n^{2} x^{2} - 4\,n^{3} - 12\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 6\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
2486918135
347112129091622754
45499209522780114310194470506
574664381267743076672435265194291602165796
611577427457265193757762562448517700444567745252488077518044
720203965157646304514803114904367700909872109509962716138122172110224803166000560
8392116005360112888814208796602312900730924170871998933966469044160503375963766853808264539520145249034544
9838060857088665308667410984491572109143244103418201847606381960148018007134345857813254949385394011387387073671425670407393764498336
1019563465222023451661239571250472502060238954763778734782263095697081052378684880579523960788922256426159724660203638899249363780487041120214277011200413488889224416
11495332425003566396980648556400693290472575143426942351465433746756432450855460240170497032616578542166374148179204883313022516039178761556813004228129608136755584984160707114136960025197804875867904
12135217535406975200517943306032013519301594913819547194541872994741477995751036814492799321950954081163798184114231089933699681653076447062545827745065231961189403775806405043481007091433804853545406829753602545610893056001661839824454306560
13395920174563270064378484164728315479841562101088008216822280560958358166172757243425337701437666368578061991322816293487071508044266833552624611723632287092607034234844614069040121996845588594952194074009018415918578609405368322256281185302086409927882482918400117926892208309814784
1412378208858377705021905837494498391851789258918563136375289325159657078444867304159355128897911162746195343137029295199135757127909048995911618558452623961449289971908577438025247550008146146541782477920444081323735398446767673631423274392671810504077457002739473913984101812606345559692804169710642825728008958871542632830511616
Technical details
Node id
n-d8c3e28cf4287787 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b4e1b99a45237f97

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:          6,           3;
  2:         48,          69,           18;
  3:        471,        1212,          909,           162;
  4:       5499,       20952,        27801,         14310,          1944;
  5:      74664,      381267,       743076,        672435,        265194,         29160;
  6:    1157742,     7457265,     19375776,      25624485,      17700444,       5677452,        524880;
  7:   20203965,   157646304,    514803114,     904367700,     909872109,     509962716,     138122172,     11022480;
  8:  392116005,  3601128888,  14208796602,   31290073092,   41708719989,   33966469044,   16050337596,   3766853808,    264539520;
  9: 8380608570, 88665308667, 410984491572, 1091432441034, 1820184760638, 1960148018007, 1343458578132, 549493853940, 113873870736, 7142567040;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(12*n^3-18*n^2+6*n)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*n)*x)*v[i - 1])/(n - 1)); 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.