Real-rooted atlas

A091044 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
23161635
34409664204
45803365122561189
561408962304256010246930
67224201676801408012288409640391
7833640322112056320798725734416384235416
89480739250688183040372736430080262144655361372105
9106601267210982451251213977602293760222822411796482621447997214
10118802059221964812812804472832974848013369344112066565242880104857646611179
111211443203241184029286401267302435094528635043847471104055050240230686724194304271669860
1213145648048732160622336032587776111132672254017536392232960403701760265289728100663296167772161583407981
1314182069888124467212446720773959683175219208890613761725825024232128512021223178241258291200436207616671088649228778026
1415224099008203673623648768171991040833495040279419289666156625921114216857613264486400109051904005888802816187904819226843545653789260175
Technical details
Node id
n-d151b46c7077d7da · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9d7293d396b320fd

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:  2,   4;
  2:  3,  16,    16;
  3:  4,  40,    96,     64;
  4:  5,  80,   336,    512,    256;
  5:  6, 140,   896,   2304,   2560,    1024;
  6:  7, 224,  2016,   7680,  14080,   12288,    4096;
  7:  8, 336,  4032,  21120,  56320,   79872,   57344,   16384;
  8:  9, 480,  7392,  50688, 183040,  372736,  430080,  262144,   65536;
  9: 10, 660, 12672, 109824, 512512, 1397760, 2293760, 2228224, 1179648, 262144;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+1+(2*n+2)*x)*P(n-1,x) + (2*x+2*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[((n + 1 + (2*n + 2)*x)*P[n - 1, x] + (2*x + 2*x^2)*D[P[n - 1, x], x])/(n)]];
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] = ((n + 1 + (2*n + 2)*x)*v[i - 1] + (2*x + 2*x^2)*deriv(v[i - 1], x))/(n)); 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

Candidate equalities ?

Same fingerprint ?

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