Real-rooted atlas

A116414 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(-5\,x - 28\,x^{2} - 54\,x^{3} - 44\,x^{4} - 13\,x^{5} + 2\,n + 21\,n x + 64\,n x^{2} + 86\,n x^{3} + 54\,n x^{4} + 13\,n x^{5}\right)\,P_{n-1}(x) + \left(-x - 13\,x^{2} - 46\,x^{3} - 70\,x^{4} - 49\,x^{5} - 13\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1156
21102233
31156995180
4120141410409981
51252381070225217605346
61303602200717911790757329133
7135507392517465444755976732585158760
8140679637036010125200260605296050140206865161
9145876966066339293475834210146565014411556032754714686
101501098139201126026060602193555526110079875906920900259575725692633
11155134519275179574114009050248381534845031790460424619003287679211168960140011740
121601617258502726551996200103874823865446010195204518563090022122352215479022048057529762992541
1316519143377039787033016501982976087171600280149831649345775105400664211335221307233782532067807654157920026
141702236431605618695213450355200451803318006848453851934131550399487938258466626405727318337335930137088973123822658542533
Technical details
Node id
n-f1d5f92ea81219e3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-1cfe01b94310c21a

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: 1,  5;
  2: 1, 10,  22;
  3: 1, 15,  69,   95;
  4: 1, 20, 141,  410,   409;
  5: 1, 25, 238, 1070,  2252,   1760;
  6: 1, 30, 360, 2200,  7179,  11790,   7573;
  7: 1, 35, 507, 3925, 17465,  44475,  59767,   32585;
  8: 1, 40, 679, 6370, 36010, 125200, 260605,  296050,  140206;
  9: 1, 45, 876, 9660, 66339, 293475, 834210, 1465650, 1441155, 603275;
%F Formula
The row polynomials satisfy (2*n+6*n*x+6*n*x^2+2*n*x^3)*P(n,x) = (2*n+(21*n-5)*x+(64*n-28)*x^2+(86*n-54)*x^3+(54*n-44)*x^4+(13*n-13)*x^5)*P(n-1,x) - (x+13*x^2+46*x^3+70*x^4+49*x^5+13*x^6)*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 + (21*n - 5)*x + (64*n - 28)*x^2 + (86*n - 54)*x^3 + (54*n - 44)*x^4 + (13*n - 13)*x^5)*P[n - 1, x] - (x + 13*x^2 + 46*x^3 + 70*x^4 + 49*x^5 + 13*x^6)*D[P[n - 1, x], x])/(2*n + 6*n*x + 6*n*x^2 + 2*n*x^3)]];
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 + (21*n - 5)*x + (64*n - 28)*x^2 + (86*n - 54)*x^3 + (54*n - 44)*x^4 + (13*n - 13)*x^5)*v[i - 1] - (x + 13*x^2 + 46*x^3 + 70*x^4 + 49*x^5 + 13*x^6)*deriv(v[i - 1], x))/(2*n + 6*n*x + 6*n*x^2 + 2*n*x^3)); 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. [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