Real-rooted atlas

A172431 → basis A013609 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(2 - 6\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-2 - 4\,x - 16\,n - 28\,n x + 16\,n^{2} + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-5\,n - 18\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2202444
37617696348
42608889603202428
584037926144416096015896
62616146883177632704156802688100152
7794053328144480200320147840537607168614836
82364418487260211210518401054080596736172032184323703748
969380619024235584049692806308800488448022149125222404608021990036
10201220201741687865922173024033445440329683202057932877045761520640112640129066412
115780646433824315617288956659216212672019446067215525171280492544254453764280320270336750467888
121647600201571201099824003522976007335392001037702400101151232067358720029652480080537600117145606389764329841776
13466483662222880373788480133409920031417344005124026880590650368048174796802734617600103966720024600576031309824149094424817611364
1413132580189678264124400640048957088001286745600023779929600316696934403062154240021352780800105064960003496796160729071616820019203440640141451734620
Technical details
Node id
n-db86dad16b91ae0d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3d61edfdb7d83d14

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:     0;
  1:     4;
  2:    20,     24;
  3:    76,    176,      96;
  4:   260,    888,     960,     320;
  5:   840,   3792,    6144,    4160,     960;
  6:  2616,  14688,   31776,   32704,   15680,    2688;
  7:  7940,  53328,  144480,  200320,  147840,   53760,    7168;
  8: 23644, 184872,  602112, 1051840, 1054080,  596736,  172032,  18432;
  9: 69380, 619024, 2355840, 4969280, 6308800, 4884480, 2214912, 522240, 46080;
%F Formula
The row polynomials satisfy (4*n^2-6*n+2)*P(n,x) = (16*n^2-16*n-2+(24*n^2-28*n-4)*x)*P(n-1,x) - (5*n+18*n*x+16*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 16*n - 2 + (24*n^2 - 28*n - 4)*x)*P[n - 1, x] - (5*n + 18*n*x + 16*n*x^2)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^2 - 16*n - 2 + (24*n^2 - 28*n - 4)*x)*v[i - 1] - (5*n + 18*n*x + 16*n*x^2)*deriv(v[i - 1], x))/(4*n^2 - 6*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023