Real-rooted atlas

A078812 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-4 + 2\,n\right)\,P_{n}(x) = \left(3 + x + 3\,n + n x\right)\,P_{n-1}(x) + \left(5 + 6\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
3628
42518346
590102364232
63004682586051091
7954190214405209064912
82939714669153420915126721468
9885025414300241896069301470168891824
10261958679612127893492435751260022122169386373
117650028736446400442322024066088620211683168270101604984
12221016928296170178017948401207430542052164682334804365330116598282
136329162939556603378072303405625180298645211052722854805049058303961226895704
1417991259156204208088942794122024706035151675206635328208612846827073260759046813108850055
15508227028127306701272081043385121034194507216335036496152135561363701214734250102960967254614437859040
Technical details
Node id
n-47d961ccf2ecbd21 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cb9ac7fd59880e43

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     0;
   2:     1;
   3:     6,     2;
   4:    25,    18,      3;
   5:    90,   102,     36,     4;
   6:   300,   468,    258,    60,     5;
   7:   954,  1902,   1440,   520,    90,     6;
   8:  2939,  7146,   6915,  3420,   915,   126,    7;
   9:  8850, 25414,  30024, 18960,  6930,  1470,  168,   8;
  10: 26195, 86796, 121278, 93492, 43575, 12600, 2212, 216, 9;
%F Formula
The row polynomials satisfy (2*n-4)*P(n,x) = (3*n+3+(n+1)*x)*P(n-1,x) + (5+6*x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 3 + (n + 1)*x)*P[n - 1, x] + (5 + 6*x + x^2)*D[P[n - 1, x], x])/(2*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((3*n + 3 + (n + 1)*x)*v[i - 1] + (5 + 6*x + x^2)*deriv(v[i - 1], x))/(2*n - 4)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023