Real-rooted atlas

A103327 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
21620541
364112567239
425657643212091393
51024281628161232220118119
640961331216640998429123641347321
71638461440921607040028800604856015275807
8655362785284874244526082393607180811424816171607521
92621441245184249036827238401770496695552160512200641140199369319
1010485765505024123863041559756812042240587059217937923294723326415402154608393
114194304241172486029312085917696768737284484300817145856420992063148852624202423318281039
12167772161048576002883584004587520004669440003175219201462272004526080091520001144000800802600251855077841
13671088644529848321358954496238760755227249868802118057984114307891242865459210998374418670080197683211793632762710812186007
142684354561946157056632501043212163481600153868042241346345369683414876163683254272115101696024938700836095488328140816889640602963018038201
Technical details
Node id
n-d007580595bb987a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-3f225bab20c1f22f

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:      4,       3;
  2:     16,      20,       5;
  3:     64,     112,      56,       7;
  4:    256,     576,     432,     120,       9;
  5:   1024,    2816,    2816,    1232,     220,     11;
  6:   4096,   13312,   16640,    9984,    2912,    364,     13;
  7:  16384,   61440,   92160,   70400,   28800,   6048,    560,    15;
  8:  65536,  278528,  487424,  452608,  239360,  71808,  11424,   816,   17;
  9: 262144, 1245184, 2490368, 2723840, 1770496, 695552, 160512, 20064, 1140, 19;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (8*n-4+(6*n-3)*x)*P(n-1,x) - (4*x+4*x^2)*d/dx 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[((8*n - 4 + (6*n - 3)*x)*P[n - 1, x] - (4*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = ((8*n - 4 + (6*n - 3)*x)*v[i - 1] - (4*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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