Real-rooted atlas

A094816 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2246
35201944
41694176106392
565492134615886854176
63262968104921808815262505652192
71957206008818119718424359115796042079751552
813700162650810924220704235489883377582176398839045412275328
91096011442952816810026024520511353746354082448818164212231284000441224463104
10986410142243328979703232633008075345601211475719281154126408739681152274322778448816604545377792
1198641011543315961072211223436275924811528134546206916945362557050887821498533456117760217853796878284547457611101008395264
12108505112182810504613834855960621625833381844571119843798860777565545025672805740144668044132136149041970947607185608492632872155899542444403165184
13130206134523478284500191976579494942359776604309579669894771761577389521204352218352414753082874424130982144224878223114104484346643585743088126683575610221181558963998919233536
141692679748632498758128028521298109481516473734071254547504595482140267400907736265443052375256374484647028880394035386516098305524590734608169738070923748640029469795281468504151551015488019699761802636225077248
15236975164805482349798299245183678318073258409981035600100877069691289728452871498643845982800354375477953158989106704011566347066619263106422262960823207310847685989051363710712588033612390203639057232587637592431682500087988693554356415402328064
Technical details
Node id
n-56185a886f282074 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b4d9f447daae5beb

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,        4;
   3:      5,       20,       19;
   4:     16,       94,      176,       106;
   5:     65,      492,     1346,      1588,       685;
   6:    326,     2968,    10492,     18088,     15262,       5056;
   7:   1957,    20600,    88181,    197184,    243591,     157960,      42079;
   8:  13700,   162650,   810924,   2207042,   3548988,    3377582,    1763988,    390454;
   9: 109601,  1442952,  8168100,  26024520,  51135374,   63540824,   48818164,  21223128,   4000441;
  10: 986410, 14224332, 89797032, 326330080, 753456012, 1147571928, 1154126408, 739681152, 274322778, 44881660;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n-2,x) + (n+(2*n+2)*x+(n+2)*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*P[n - 2, x] + (n + (2*n + 2)*x + (n + 2)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*v[i - 2] + (n + (2*n + 2)*x + (n + 2)*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, v[i])))};
T(10)

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. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023