Real-rooted atlas

A232535 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(24 + 123\,x + 249\,x^{2} + 249\,x^{3} + 123\,x^{4} + 24\,x^{5} - 43\,n - 213\,n x - 418\,n x^{2} - 405\,n x^{3} - 193\,n x^{4} - 36\,n x^{5} + 22\,n^{2} + 101\,n^{2} x + 182\,n^{2} x^{2} + 161\,n^{2} x^{3} + 70\,n^{2} x^{4} + 12\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(18 + 135\,x + 408\,x^{2} + 642\,x^{3} + 558\,x^{4} + 255\,x^{5} + 48\,x^{6} - 12\,n - 86\,n x - 250\,n x^{2} - 378\,n x^{3} - 314\,n x^{4} - 136\,n x^{5} - 24\,n x^{6}\right)\,P_{n-1}'(x) + \left(6\,x + 42\,x^{2} + 120\,x^{3} + 180\,x^{4} + 150\,x^{5} + 66\,x^{6} + 12\,x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1358
212382979
348224341169782
41921184269626849857741
57685888178402668819703574176628
630722816010649621260823624413849833461758539
7122881310725941761483520220278419447689451291950257508762

All 15 rows

Technical details
Node id
n-cf40632e68daf3ea · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-76acbb44bc26a5c0

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:      3,       5;
  2:     12,      38,       29;
  3:     48,     224,      341,      169;
  4:    192,    1184,     2696,     2684,       985;
  5:    768,    5888,    17840,    26688,     19703,      5741;
  6:   3072,   28160,   106496,   212608,    236244,    138498,     33461;
  7:  12288,  131072,   594176,  1483520,   2202784,   1944768,    945129,    195025;
  8:  49152,  598016,  3160064,  9470976,  17605760,  20783168,  15212880,   6312376,  1136689;
  9: 196608, 2686976, 16216064, 56713216, 126654720, 187283712, 183347744, 114582016, 41475883, 6625109;
%F Formula
The row polynomials satisfy (n^2+2*n^2*x+n^2*x^2)*P(n,x) = (22*n^2-43*n+24+(101*n^2-213*n+123)*x+(182*n^2-418*n+249)*x^2+(161*n^2-405*n+249)*x^3+(70*n^2-193*n+123)*x^4+(12*n^2-36*n+24)*x^5)*P(n-1,x) - (12*n-18+(86*n-135)*x+(250*n-408)*x^2+(378*n-642)*x^3+(314*n-558)*x^4+(136*n-255)*x^5+(24*n-48)*x^6)*d/dx P(n-1,x) + (6*x+42*x^2+120*x^3+180*x^4+150*x^5+66*x^6+12*x^7)*d^2/dx^2 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[((22*n^2 - 43*n + 24 + (101*n^2 - 213*n + 123)*x + (182*n^2 - 418*n + 249)*x^2 + (161*n^2 - 405*n + 249)*x^3 + (70*n^2 - 193*n + 123)*x^4 + (12*n^2 - 36*n + 24)*x^5)*P[n - 1, x] - (12*n - 18 + (86*n - 135)*x + (250*n - 408)*x^2 + (378*n - 642)*x^3 + (314*n - 558)*x^4 + (136*n - 255)*x^5 + (24*n - 48)*x^6)*D[P[n - 1, x], x] + (6*x + 42*x^2 + 120*x^3 + 180*x^4 + 150*x^5 + 66*x^6 + 12*x^7)*D[P[n - 1, x], {x, 2}])/(n^2 + 2*n^2*x + n^2*x^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] = ((22*n^2 - 43*n + 24 + (101*n^2 - 213*n + 123)*x + (182*n^2 - 418*n + 249)*x^2 + (161*n^2 - 405*n + 249)*x^3 + (70*n^2 - 193*n + 123)*x^4 + (12*n^2 - 36*n + 24)*x^5)*v[i - 1] - (12*n - 18 + (86*n - 135)*x + (250*n - 408)*x^2 + (378*n - 642)*x^3 + (314*n - 558)*x^4 + (136*n - 255)*x^5 + (24*n - 48)*x^6)*deriv(v[i - 1], x) + (6*x + 42*x^2 + 120*x^3 + 180*x^4 + 150*x^5 + 66*x^6 + 12*x^7)*deriv(deriv(v[i - 1], x), x))/(n^2 + 2*n^2*x + n^2*x^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

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