Real-rooted atlas

A232535 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-12 - 21\,x - 9\,x^{2} + 17\,n + 31\,n x + 15\,n x^{2} - 2\,n^{2} - 9\,n^{2} x - 6\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-18 - 57\,x - 60\,x^{2} - 21\,x^{3} + 12\,n + 34\,n x + 34\,n x^{2} + 12\,n x^{3}\right)\,P_{n-1}'(x) + \left(-6\,x - 18\,x^{2} - 18\,x^{3} - 6\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
21210123
34864211134
4192352200361781
5768179214564805514552
6307287049216448098078126531
7122884096053504345601142417921051154634

All 15 rows

Technical details
Node id
n-a81fc29c539209bb · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-3908c67b5f7a1c29

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:     12,     10,       1;
  3:     48,     64,      21,       1;
  4:    192,    352,     200,      36,      1;
  5:    768,   1792,    1456,     480,     55,      1;
  6:   3072,   8704,    9216,    4480,    980,     78,     1;
  7:  12288,  40960,   53504,   34560,  11424,   1792,   105,    1;
  8:  49152, 188416,  292864,  236544, 105600,  25536,  3024,  136,   1;
  9: 196608, 851968, 1536000, 1490944, 841984, 278784, 51744, 4800, 171, 1;
%F Formula
The row polynomials satisfy n^2*P(n,x) = -(2*n^2-17*n+12+(9*n^2-31*n+21)*x+(6*n^2-15*n+9)*x^2)*P(n-1,x) + (12*n-18+(34*n-57)*x+(34*n-60)*x^2+(12*n-21)*x^3)*d/dx P(n-1,x) - (6*x+18*x^2+18*x^3+6*x^4)*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[(-(2*n^2 - 17*n + 12 + (9*n^2 - 31*n + 21)*x + (6*n^2 - 15*n + 9)*x^2)*P[n - 1, x] + (12*n - 18 + (34*n - 57)*x + (34*n - 60)*x^2 + (12*n - 21)*x^3)*D[P[n - 1, x], x] - (6*x + 18*x^2 + 18*x^3 + 6*x^4)*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] = (-(2*n^2 - 17*n + 12 + (9*n^2 - 31*n + 21)*x + (6*n^2 - 15*n + 9)*x^2)*v[i - 1] + (12*n - 18 + (34*n - 57)*x + (34*n - 60)*x^2 + (12*n - 21)*x^3)*deriv(v[i - 1], x) - (6*x + 18*x^2 + 18*x^3 + 6*x^4)*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

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