Real-rooted atlas

A033282 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(2 - 4\,x^{2} + 4\,n + 9\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 6\,x^{2} - 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
25151131
314639345215
4422525605461971597
513299029404320314090312425
6429386114355282153091517901427999955
71430150156706716516524216521136510167520793824675
84862583443043049009001655654193393214020165767081030496939769
916796226746135252046781281033905615140034146884929103536327043851885959334605
10587868817905920590234304206051864010659448812965316010752651058188585185532752646723513972967
1120801234321982561713011415789033746394069479625610165903441056991320765316860367494435105324681136488694501041935
127429001337220010983033654424854018121365904270848120730525224091372358888293857660532790676022991601605984077507103937339784038517
1326744405215158046743268025496328009442140136250692645064909151676071786574288783856890966312170264436435419592142757372163402883848372693519354455513105
14969484520359174519774140751177321918548002149195141775518885312798513027523691390079668552103219647614352385468563894799245538217197880917074311936798863319688015193179928556219
Technical details
Node id
n-c0e24188ccfeaae9 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ca6d936c14642db2

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,      3;
  2:     5,     15,      11;
  3:    14,     63,      93,      45;
  4:    42,    252,     560,     546,      197;
  5:   132,    990,    2940,    4320,     3140,      903;
  6:   429,   3861,   14355,   28215,    30915,    17901,     4279;
  7:  1430,  15015,   67067,  165165,   242165,   211365,   101675,   20793;
  8:  4862,  58344,  304304,  900900,  1655654,  1933932,  1402016,  576708,  103049;
  9: 16796, 226746, 1352520, 4678128, 10339056, 15140034, 14688492, 9103536, 3270438, 518859;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (4*n+2+9*n*x+(4*n-4)*x^2)*P(n-1,x) - (2*x+6*x^2+4*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[((4*n + 2 + 9*n*x + (4*n - 4)*x^2)*P[n - 1, x] - (2*x + 6*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n + 2)]];
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] = ((4*n + 2 + 9*n*x + (4*n - 4)*x^2)*v[i - 1] - (2*x + 6*x^2 + 4*x^3)*deriv(v[i - 1], x))/(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

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