Real-rooted atlas

A207824 → basis A013609 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−1/2, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-3\,x - 8\,x^{2} + 2\,n + 9\,n x + 8\,n x^{2}\right)\,P_{n-1}(x) + \left(-x - 6\,x^{2} - 8\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1268
24243563
3872212204496
416192852165611893905
5324802848835212102693030744
6641152856033600734648481640391242047
71282688240001180803456806020645775762354161905632
82566144640643790081391280324460846940723851472137210515003009
95121382416486411397125032608147188162851011235265312252754667997214118118440
101024307204124163262464168394245925427214394184023834668825744873216379575246611179929944511
11204867584100864089856005308876821840537663837670413256256001916467480183701858410507214042716698607321437648
124096147456242176023992320159678720752080896257034624064222571521164274672014934495360128661055806683872392158340798157641556673
1381923194885726208624476164623027202453248512960008908828048186368611806596489839984544011342271584088727359680422193289109228778026453811015736
1416384688128133652481591050241296869376765632102433759990528112944826368288080091264557461391616805601757600843056519040603936710320265089657120537892601753572846569215
Technical details
Node id
n-d0cb59aeab637630 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-611b245e25712c49

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:   2,     6;
  2:   4,    24,     35;
  3:   8,    72,    212,     204;
  4:  16,   192,    852,    1656,    1189;
  5:  32,   480,   2848,    8352,   12102,     6930;
  6:  64,  1152,   8560,   33600,   73464,    84816,    40391;
  7: 128,  2688,  24000,  118080,  345680,   602064,   577576,   235416;
  8: 256,  6144,  64064,  379008, 1391280,  3244608,  4694072,  3851472,  1372105;
  9: 512, 13824, 164864, 1139712, 5032608, 14718816, 28510112, 35265312, 25275466, 7997214;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+(9*n-3)*x+(8*n-8)*x^2)*P(n-1,x) - (x+6*x^2+8*x^3)*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[((2*n + (9*n - 3)*x + (8*n - 8)*x^2)*P[n - 1, x] - (x + 6*x^2 + 8*x^3)*D[P[n - 1, x], x])/(n)]];
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 + (9*n - 3)*x + (8*n - 8)*x^2)*v[i - 1] - (x + 6*x^2 + 8*x^3)*deriv(v[i - 1], x))/(n)); 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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023