Real-rooted atlas

A063007 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-6 - 15\,x - 8\,x^{2} + 4\,n + 9\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 6\,x^{2} - 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2235
36181337
4209013263305
5704209309003212641
62521890560081905910168323525
79248316308706048065940379268989213445
8343236036160776395010577080501228239624486391961825
9128701544408048042378376435897050727603660300149709626572918205345
10486206563703912480135135002980177243513470420604802595186092744101462563170195525
11184756277134018597150735134401895493603330807484039335303337963201798740905707449080974531600472677
1270543211639628868353203866019301141208640234507873634228434243548374830256029774012245161503493674364504671915122515985
1327041564867480839962722819787367606580546830154840308482643134894432978129184298473575401910971062082153251182129223564251595969143457011569
1410400600202811700181719283299053234283664542882097161794730189936558240277162821936301896418824242419757580139482383040544551052501292916588614099336191365435096485
15401166008424486008180071900486748080001982849428565849495051401288652314950215359722864027434991183602651111511048191285952858099930160512035730280404078265638990792384825313033485491077
Technical details
Node id
n-a882d0a2ea16bc46 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c277ed2a39f7c92c

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     2,      3;
   3:     6,     18,      13;
   4:    20,     90,     132,       63;
   5:    70,    420,     930,      900,      321;
   6:   252,   1890,    5600,     8190,     5910,     1683;
   7:   924,   8316,   30870,    60480,    65940,    37926,     8989;
   8:  3432,  36036,  160776,   395010,   577080,   501228,   239624,    48639;
   9: 12870, 154440,  804804,  2378376,  4358970,  5072760,  3660300,  1497096,  265729;
  10: 48620, 656370, 3912480, 13513500, 29801772, 43513470, 42060480, 25951860, 9274410, 1462563;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n-6+(9*n-15)*x+(4*n-8)*x^2)*P(n-1,x) - (2*x+6*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 6 + (9*n - 15)*x + (4*n - 8)*x^2)*P[n - 1, x] - (2*x + 6*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n - 6 + (9*n - 15)*x + (4*n - 8)*x^2)*v[i - 1] - (2*x + 6*x^2 + 4*x^3)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Candidate equalities ?

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