Real-rooted atlas

A069466 → basis A007318 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(8 + 4\,x - 30\,n - 14\,n x + 28\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(8 + 12\,x + 4\,x^{2} - 16\,n - 24\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
278486132
312601140300202720
422470268801092016807062020
54241166312603477608568088202521497888
68305836147913921029798035112005959804435292437547664
71669290483461137682913870961272070803039036038198162162163432966066816
8341993223080918323207992784800425801376013171158002363961602306304010296001287025340180580
97110981306018906650334021435818976013488899424051324633360120439519201698977280133024320481338048620674628949280
1014960530264684415675790240565151329314041032681747201855279105680539333803008100333593360114844329607399477802217072018475618173703522832
113177739707760810309639874133614702002487172012099684296700063417814780320220256215058885108306819616777061565280739116378003996272280100876776705432494297477821056
12680354749147164240650431383398437836547182514083481216102684320207600453354294083935461952742423371280450918444577167963328567414405558045689419776021059966928454298208270415613551531564682384
1314664155597771400561629400570966009651691172214000098180013091960800656564770431570003030717085135740098620612586496002268408302160000363940322545800394651287030002730656728800108707071200202811700010400600374024384777859200
14317877850826299800131053857846196320024438907633633038002723634969142651200201891960581891580010480344051104928003905906086144062001053685625417568002047016738386620028112914157328002634682176126001585825244640055124220060089861184004011660010382416689417883200

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(n^{2}\right)\,P_{n}(x) = \left(-12 - 24\,x - 12\,x^{2} + 32\,n + 64\,n x + 32\,n x^{2} - 16\,n^{2} - 32\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(6 + 2\,x - 24\,n - 8\,n x + 24\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 2\,x\)
Technical details
Node id
n-c73008ca3f0e3c9c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-f042c28306dcf91e)
Fingerprint
f6-c880512e8efe320e

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,            2;
  2:          78,           48,            6;
  3:        1260,         1140,          300,           20;
  4:       22470,        26880,        10920,         1680,          70;
  5:      424116,       631260,       347760,        85680,        8820,         252;
  6:     8305836,     14791392,     10297980,      3511200,      595980,       44352,        924;
  7:   166929048,    346113768,    291387096,    127207080,    30390360,     3819816,     216216,      3432;
  8:  3419932230,   8091832320,   7992784800,   4258013760,  1317115800,   236396160,   23063040,   1029600,   12870;
  9: 71109813060, 189066503340, 214358189760, 134888994240, 51324633360, 12043951920, 1698977280, 133024320, 4813380, 48620;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (28*n^2-30*n+8+(12*n^2-14*n+4)*x)*P(n-1,x) - (16*n-8+(24*n-12)*x+(8*n-4)*x^2)*d/dx 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[((28*n^2 - 30*n + 8 + (12*n^2 - 14*n + 4)*x)*P[n - 1, x] - (16*n - 8 + (24*n - 12)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(n^2)]];
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] = ((28*n^2 - 30*n + 8 + (12*n^2 - 14*n + 4)*x)*v[i - 1] - (16*n - 8 + (24*n - 12)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(n^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023