Real-rooted atlas

A207607 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(8 + 32\,x + 48\,x^{2} + 32\,x^{3} + 8\,x^{4} - 16\,n - 64\,n x - 96\,n x^{2} - 64\,n x^{3} - 16\,n x^{4} + 8\,n^{2} + 32\,n^{2} x + 48\,n^{2} x^{2} + 32\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n}(x) = \left(8 + 74\,x + 260\,x^{2} + 490\,x^{3} + 560\,x^{4} + 398\,x^{5} + 164\,x^{6} + 30\,x^{7} - 16\,n - 142\,n x - 490\,n x^{2} - 905\,n x^{3} - 1000\,n x^{4} - 676\,n x^{5} - 262\,n x^{6} - 45\,n x^{7} + 8\,n^{2} + 68\,n^{2} x + 230\,n^{2} x^{2} + 415\,n^{2} x^{3} + 440\,n^{2} x^{4} + 278\,n^{2} x^{5} + 98\,n^{2} x^{6} + 15\,n^{2} x^{7}\right)\,P_{n-1}(x) + \left(4\,x + 46\,x^{2} + 224\,x^{3} + 576\,x^{4} + 844\,x^{5} + 710\,x^{6} + 320\,x^{7} + 60\,x^{8} - 4\,n x - 46\,n x^{2} - 200\,n x^{3} - 450\,n x^{4} - 580\,n x^{5} - 434\,n x^{6} - 176\,n x^{7} - 30\,n x^{8}\right)\,P_{n-1}'(x) + \left(3\,x^{3} + 30\,x^{4} + 105\,x^{5} + 180\,x^{6} + 165\,x^{7} + 78\,x^{8} + 15\,x^{9}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
229819
3215332171
42217611155265
5227137324355144989
6233216714125110983773691
7239313133532564527331698713775

All 15 rows

Technical details
Node id
n-5eae171d920f870a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2b83e2879aa49b31

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: 2,  9,   8;
  3: 2, 15,  33,   21;
  4: 2, 21,  76,  111,    55;
  5: 2, 27, 137,  324,   355,   144;
  6: 2, 33, 216,  714,  1251,  1098,   377;
  7: 2, 39, 313, 1335,  3256,  4527,  3316,   987;
  8: 2, 45, 428, 2241,  7045, 13581, 15646,  9837,  2584;
  9: 2, 51, 561, 3486, 13455, 33381, 53133, 52248, 28779, 6765;
%F Formula
The row polynomials satisfy (8*n^2-16*n+8+(32*n^2-64*n+32)*x+(48*n^2-96*n+48)*x^2+(32*n^2-64*n+32)*x^3+(8*n^2-16*n+8)*x^4)*P(n,x) = (8*n^2-16*n+8+(68*n^2-142*n+74)*x+(230*n^2-490*n+260)*x^2+(415*n^2-905*n+490)*x^3+(440*n^2-1000*n+560)*x^4+(278*n^2-676*n+398)*x^5+(98*n^2-262*n+164)*x^6+(15*n^2-45*n+30)*x^7)*P(n-1,x) - ((4*n-4)*x+(46*n-46)*x^2+(200*n-224)*x^3+(450*n-576)*x^4+(580*n-844)*x^5+(434*n-710)*x^6+(176*n-320)*x^7+(30*n-60)*x^8)*d/dx P(n-1,x) + (3*x^3+30*x^4+105*x^5+180*x^6+165*x^7+78*x^8+15*x^9)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 16*n + 8 + (68*n^2 - 142*n + 74)*x + (230*n^2 - 490*n + 260)*x^2 + (415*n^2 - 905*n + 490)*x^3 + (440*n^2 - 1000*n + 560)*x^4 + (278*n^2 - 676*n + 398)*x^5 + (98*n^2 - 262*n + 164)*x^6 + (15*n^2 - 45*n + 30)*x^7)*P[n - 1, x] - ((4*n - 4)*x + (46*n - 46)*x^2 + (200*n - 224)*x^3 + (450*n - 576)*x^4 + (580*n - 844)*x^5 + (434*n - 710)*x^6 + (176*n - 320)*x^7 + (30*n - 60)*x^8)*D[P[n - 1, x], x] + (3*x^3 + 30*x^4 + 105*x^5 + 180*x^6 + 165*x^7 + 78*x^8 + 15*x^9)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 16*n + 8 + (32*n^2 - 64*n + 32)*x + (48*n^2 - 96*n + 48)*x^2 + (32*n^2 - 64*n + 32)*x^3 + (8*n^2 - 16*n + 8)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 16*n + 8 + (68*n^2 - 142*n + 74)*x + (230*n^2 - 490*n + 260)*x^2 + (415*n^2 - 905*n + 490)*x^3 + (440*n^2 - 1000*n + 560)*x^4 + (278*n^2 - 676*n + 398)*x^5 + (98*n^2 - 262*n + 164)*x^6 + (15*n^2 - 45*n + 30)*x^7)*v[i - 1] - ((4*n - 4)*x + (46*n - 46)*x^2 + (200*n - 224)*x^3 + (450*n - 576)*x^4 + (580*n - 844)*x^5 + (434*n - 710)*x^6 + (176*n - 320)*x^7 + (30*n - 60)*x^8)*deriv(v[i - 1], x) + (3*x^3 + 30*x^4 + 105*x^5 + 180*x^6 + 165*x^7 + 78*x^8 + 15*x^9)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 16*n + 8 + (32*n^2 - 64*n + 32)*x + (48*n^2 - 96*n + 48)*x^2 + (32*n^2 - 64*n + 32)*x^3 + (8*n^2 - 16*n + 8)*x^4)); 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