Real-rooted atlas

A207607 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(8 - 16\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(56 + 66\,x + 18\,x^{2} - 100\,n - 114\,n x - 30\,n x^{2} + 44\,n^{2} + 48\,n^{2} x + 12\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(68 + 174\,x + 148\,x^{2} + 42\,x^{3} - 50\,n - 120\,n x - 94\,n x^{2} - 24\,n x^{3}\right)\,P_{n-1}'(x) + \left(15 + 57\,x + 81\,x^{2} + 51\,x^{3} + 12\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
287116
3213012164
45510973181256
51443653441452511024
6377116414168502553314096
7987359353584184180641342116384

All 15 rows

Technical details
Node id
n-3dd3bd37c05e36d2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-429c048c3a049f47

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:    3,     1;
  2:    8,     7,     1;
  3:   21,    30,    12,     1;
  4:   55,   109,    73,    18,     1;
  5:  144,   365,   344,   145,    25,     1;
  6:  377,  1164,  1416,   850,   255,    33,    1;
  7:  987,  3593,  5358,  4184,  1806,   413,   42,   1;
  8: 2584, 10835, 19139, 18422, 10415,  3458,  630,  52,  1;
  9: 6765, 32106, 65556, 75051, 52557, 22995, 6132, 918, 63, 1;
%F Formula
The row polynomials satisfy (8*n^2-16*n+8)*P(n,x) = (44*n^2-100*n+56+(48*n^2-114*n+66)*x+(12*n^2-30*n+18)*x^2)*P(n-1,x) - (50*n-68+(120*n-174)*x+(94*n-148)*x^2+(24*n-42)*x^3)*d/dx P(n-1,x) + (15+57*x+81*x^2+51*x^3+12*x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((44*n^2 - 100*n + 56 + (48*n^2 - 114*n + 66)*x + (12*n^2 - 30*n + 18)*x^2)*P[n - 1, x] - (50*n - 68 + (120*n - 174)*x + (94*n - 148)*x^2 + (24*n - 42)*x^3)*D[P[n - 1, x], x] + (15 + 57*x + 81*x^2 + 51*x^3 + 12*x^4)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 16*n + 8)]];
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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = ((44*n^2 - 100*n + 56 + (48*n^2 - 114*n + 66)*x + (12*n^2 - 30*n + 18)*x^2)*v[i - 1] - (50*n - 68 + (120*n - 174)*x + (94*n - 148)*x^2 + (24*n - 42)*x^3)*deriv(v[i - 1], x) + (15 + 57*x + 81*x^2 + 51*x^3 + 12*x^4)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 16*n + 8)); 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. [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