Real-rooted atlas

A123519 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 2\,n\right)\,P_{n}(x) = \left(13 + 7\,x + 10\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-6 - 10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 11 + 8\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0118120
14147151104
2153240126241544
357111378342703512848
4213151444851223250548114912
57953225592601315555507586163178080
629681967361319009716841590103041372801408832
7110771407841641988562332298498978301923620769912140672

All 15 rows

Technical details
Node id
n-3ad802ff9e828000 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-935c57c7e92571a4

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      11,       8,        1;
  1:      41,      47,       15,        1;
  2:     153,     240,      126,       24,        1;
  3:     571,    1137,      834,      270,       35,       1;
  4:    2131,    5144,     4851,     2232,      505,      48,       1;
  5:    7953,   22559,    26013,    15555,     5075,     861,      63,      1;
  6:   29681,   96736,   131900,    97168,    41590,   10304,    1372,     80,     1;
  7:  110771,  407841,   641988,   562332,   298498,   97830,   19236,   2076,    99,    1;
  8:  413403, 1696808,  3028581,  3075600,  1950830,  797664,  209034,  33648,  3015,  120,   1;
  9: 1542841, 6984655, 13939499, 16105413, 11889370, 5835158, 1918070, 414282, 55869, 4235, 143, 1;
%F Formula
The row polynomials satisfy (2*n+3)*P(n,x) = (10*n+13+(6*n+7)*x)*P(n-1,x) - (6+10*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 8*x + x^2.
%t Mathematica
P[0, x_] := 11 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((10*n + 13 + (6*n + 7)*x)*P[n - 1, x] - (6 + 10*x + 4*x^2)*D[P[n - 1, x], x])/(2*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 11 + 8*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((10*n + 13 + (6*n + 7)*x)*v[i - 1] - (6 + 10*x + 4*x^2)*deriv(v[i - 1], x))/(2*n + 3)); 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