Real-rooted atlas

chebyshev_t → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−3, −1)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
278217
3264524499
497224184648577
536210451160620160163363
6135146806546473618723843219601
7504220377343843141616800526489664114243
8188178691217174418969612816054272140802048128665857
97022636490582641610708568752324681441639683628846082563880899
10262087151316038637305746560551448035687681578080471040908801024051222619537
119781226211909176548242964284432675808248358881328588850040321300992222464225281024131836323
123650401252907207917938414813446418447444016112640010123724846116864151211523481600534528491522048768398401
13136234821022517733496441527212537801001490880988618592713915904381937920151417344439462409078784126464010649640964478554083
14508435274109633361523900770343616896052639966405796329728473166355229094666241354424064475138048123673088231669762953216229376819226102926097
Technical details
Node id
n-e77cd13e9e0a7e61 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-3661b733ab1f9507

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,      1;
  2:     7,      8,      2;
  3:    26,     45,     24,       4;
  4:    97,    224,    184,      64,      8;
  5:   362,   1045,   1160,     620,    160,     16;
  6:  1351,   4680,   6546,    4736,   1872,    384,     32;
  7:  5042,  20377,  34384,   31416,  16800,   5264,    896,    64;
  8: 18817,  86912, 171744,  189696, 128160,  54272,  14080,  2048,  128;
  9: 70226, 364905, 826416, 1070856, 875232, 468144, 163968, 36288, 4608, 256;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) + (3+4*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x] + (3 + 4*x + x^2)*D[P[n - 1, x], 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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1] + (3 + 4*x + x^2)*deriv(v[i - 1], x))/(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

Same fingerprint ?

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