Real-rooted atlas

A207606 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −1/2)coefficients of one signHadamard kernel

Recurrence

\(\left(18 + 36\,x + 18\,x^{2} + 24\,n + 48\,n x + 24\,n x^{2} + 8\,n^{2} + 16\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-33 - 189\,x - 387\,x^{2} - 339\,x^{3} - 108\,x^{4} - 34\,n - 186\,n x - 342\,n x^{2} - 226\,n x^{3} + 36\,n x^{5} + 4\,n^{2} + 20\,n^{2} x + 64\,n^{2} x^{2} + 120\,n^{2} x^{3} + 108\,n^{2} x^{4} + 36\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(57 + 330\,x + 756\,x^{2} + 858\,x^{3} + 483\,x^{4} + 108\,x^{5} - 10\,n - 92\,n x - 336\,n x^{2} - 628\,n x^{3} - 638\,n x^{4} - 336\,n x^{5} - 72\,n x^{6}\right)\,P_{n-1}'(x) + \left(15 + 123\,x + 426\,x^{2} + 810\,x^{3} + 915\,x^{4} + 615\,x^{5} + 228\,x^{6} + 36\,x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 13 + 37\,x + 26\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
013372676
13414620797484
289511109210303623084
32331675478768014805135119652
46105268188583582638105215195042125228
5159716103692761648602344251992159369518817797988
641814820924223669287412341001401790991760399628702265085004
71094614205481646527280515840375830884578558904760502167786126208732403076
82865741337526747761022445825571245437251715177361041920128222143496957625978122206481516
97502511908078566309368694431054986252107428472999075593040736972152688041013519602856283136504011315758308
101964183401832269319721288861804152787109491309221577898868192265880817042729851071824645453988612116288658136234828384382092
1151422996503118338029643919356015735770304049898066770332841410964940234116801304309196602610520272886120026923994701614455084352753427641412
12134626927213181254810078146494990857775222881653542779635418138084576805468927177266275767907300127480926861362472290211487210645211889640739189750626340455961516
13352457876345850770087811479848557320657022050650801268541550194238602842926978844047268365274472907477053806311944052449346981551153765269153755932148575559663357081589772169481164964
1492274652132301432305064124154740247187220277626024831304648265111226303213279035382362128799406492269164641669026753766248882068513297572121960881848053012940022016021508832030079200280264288528213824544308684
Technical details
Node id
n-9099ed814bab6006 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-dfae2fca4d54dfc5

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    13,      37,      26;
  1:    34,     146,     207,       97;
  2:    89,     511,    1092,     1030,       362;
  3:   233,    1675,    4787,     6801,      4805,      1351;
  4:   610,    5268,   18858,    35826,     38105,     21519,      5042;
  5:  1597,   16103,   69276,   164860,    234425,    199215,     93695,     18817;
  6:  4181,   48209,  242236,   692874,   1234100,   1401790,    991760,    399628,     70226;
  7: 10946,  142054,  816465,  2728051,   5840375,   8308845,   7855890,   4760502,   1677861,    262087;
  8: 28657,  413375, 2674776, 10224458,  25571245,  43725171,  51773610,  41920128,  22214349,   6957625,   978122;
  9: 75025, 1190807, 8566309, 36869443, 105498625, 210742847, 299907559, 304073697, 215268804, 101351960, 28562831, 3650401;
%F Formula
The row polynomials satisfy (8*n^2+24*n+18+(16*n^2+48*n+36)*x+(8*n^2+24*n+18)*x^2)*P(n,x) = (4*n^2-34*n-33+(20*n^2-186*n-189)*x+(64*n^2-342*n-387)*x^2+(120*n^2-226*n-339)*x^3+(108*n^2-108)*x^4+(36*n^2+36*n)*x^5)*P(n-1,x) - (10*n-57+(92*n-330)*x+(336*n-756)*x^2+(628*n-858)*x^3+(638*n-483)*x^4+(336*n-108)*x^5+72*n*x^6)*d/dx P(n-1,x) + (15+123*x+426*x^2+810*x^3+915*x^4+615*x^5+228*x^6+36*x^7)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 13 + 37*x + 26*x^2.
%t Mathematica
P[0, x_] := 13 + 37*x + 26*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 34*n - 33 + (20*n^2 - 186*n - 189)*x + (64*n^2 - 342*n - 387)*x^2 + (120*n^2 - 226*n - 339)*x^3 + (108*n^2 - 108)*x^4 + (36*n^2 + 36*n)*x^5)*P[n - 1, x] - (10*n - 57 + (92*n - 330)*x + (336*n - 756)*x^2 + (628*n - 858)*x^3 + (638*n - 483)*x^4 + (336*n - 108)*x^5 + 72*n*x^6)*D[P[n - 1, x], x] + (15 + 123*x + 426*x^2 + 810*x^3 + 915*x^4 + 615*x^5 + 228*x^6 + 36*x^7)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18 + (16*n^2 + 48*n + 36)*x + (8*n^2 + 24*n + 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 13 + 37*x + 26*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 - 34*n - 33 + (20*n^2 - 186*n - 189)*x + (64*n^2 - 342*n - 387)*x^2 + (120*n^2 - 226*n - 339)*x^3 + (108*n^2 - 108)*x^4 + (36*n^2 + 36*n)*x^5)*v[i - 1] - (10*n - 57 + (92*n - 330)*x + (336*n - 756)*x^2 + (628*n - 858)*x^3 + (638*n - 483)*x^4 + (336*n - 108)*x^5 + 72*n*x^6)*deriv(v[i - 1], x) + (15 + 123*x + 426*x^2 + 810*x^3 + 915*x^4 + 615*x^5 + 228*x^6 + 36*x^7)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18 + (16*n^2 + 48*n + 36)*x + (8*n^2 + 24*n + 18)*x^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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023