Real-rooted atlas

A232535 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(6 - 14\,n - n x + 13\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(39 + 21\,x - 28\,n - 18\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(12 + 18\,x + 6\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
22920352
3169166494388
498512565549652896
55741900252861416165621616
63346162268456691660830712607161344
719502542004636951916999643359595038581204288

All 15 rows

Technical details
Node id
n-acd4a0edde428489 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-12c19fdbc0f3a4d9

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:       5,        2;
  2:      29,       20,        3;
  3:     169,      166,       49,        4;
  4:     985,     1256,      554,       96,       5;
  5:    5741,     9002,     5286,     1416,     165,       6;
  6:   33461,    62268,    45669,    16608,    3071,     260,      7;
  7:  195025,   420046,   369519,   169996,   43359,    5950,    385,     8;
  8: 1136689,  2781136,  2853540,  1588800,  518190,   99408,  10612,   544,   9;
  9: 6625109, 18150098, 21278876, 13910800, 5533870, 1370164, 206844, 17760, 741, 10;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (13*n^2-14*n+6+(3*n^2-n)*x)*P(n-1,x) - (28*n-39+(18*n-21)*x+2*n*x^2)*d/dx P(n-1,x) + (12+18*x+6*x^2)*d^2/dx^2 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[((13*n^2 - 14*n + 6 + (3*n^2 - n)*x)*P[n - 1, x] - (28*n - 39 + (18*n - 21)*x + 2*n*x^2)*D[P[n - 1, x], x] + (12 + 18*x + 6*x^2)*D[P[n - 1, x], {x, 2}])/(n^2)]];
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] = ((13*n^2 - 14*n + 6 + (3*n^2 - n)*x)*v[i - 1] - (28*n - 39 + (18*n - 21)*x + 2*n*x^2)*deriv(v[i - 1], x) + (12 + 18*x + 6*x^2)*deriv(deriv(v[i - 1], x), x))/(n^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

Candidate equalities ?

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