Real-rooted atlas

A207607 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −1/2)coefficients of one signHadamard kernel

Recurrence

\(\left(8 + 16\,x + 8\,x^{2} - 16\,n - 32\,n x - 16\,n x^{2} + 8\,n^{2} + 16\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n}(x) = \left(40 + 194\,x + 406\,x^{2} + 462\,x^{3} + 282\,x^{4} + 72\,x^{5} - 44\,n - 214\,n x - 470\,n x^{2} - 582\,n x^{3} - 390\,n x^{4} - 108\,n x^{5} + 4\,n^{2} + 20\,n^{2} x + 64\,n^{2} x^{2} + 120\,n^{2} x^{3} + 108\,n^{2} x^{4} + 36\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(52 + 374\,x + 1116\,x^{2} + 1768\,x^{3} + 1568\,x^{4} + 738\,x^{5} + 144\,x^{6} - 10\,n - 92\,n x - 336\,n x^{2} - 628\,n x^{3} - 638\,n x^{4} - 336\,n x^{5} - 72\,n x^{6}\right)\,P_{n-1}'(x) + \left(15 + 123\,x + 426\,x^{2} + 810\,x^{3} + 915\,x^{4} + 615\,x^{5} + 228\,x^{6} + 36\,x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1358
28251952
3219614471332
4553317397262652116
5144107531844679341398913484
6377336012396242382649915363369185924
798710225451501101761604941395916713413775547532

All 15 rows

Technical details
Node id
n-65cabe4159f82979 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-914046bce8f464f8

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:    8,    25,     19;
  3:   21,    96,    144,      71;
  4:   55,   331,    739,     726,     265;
  5:  144,  1075,   3184,    4679,    3413,     989;
  6:  377,  3360,  12396,   24238,   26499,   15363,    3691;
  7:  987, 10225,  45150,  110176,  160494,  139591,   67134,   13775;
  8: 2584, 30509, 156857,  458738,  834815,  968150,  698850,  287116,   51409;
  9: 6765, 89664, 526020, 1792941, 3913401, 5672955, 5462340, 3369114, 1208007, 191861;
%F Formula
The row polynomials satisfy (8*n^2-16*n+8+(16*n^2-32*n+16)*x+(8*n^2-16*n+8)*x^2)*P(n,x) = (4*n^2-44*n+40+(20*n^2-214*n+194)*x+(64*n^2-470*n+406)*x^2+(120*n^2-582*n+462)*x^3+(108*n^2-390*n+282)*x^4+(36*n^2-108*n+72)*x^5)*P(n-1,x) - (10*n-52+(92*n-374)*x+(336*n-1116)*x^2+(628*n-1768)*x^3+(638*n-1568)*x^4+(336*n-738)*x^5+(72*n-144)*x^6)*d/dx P(n-1,x) + (15+123*x+426*x^2+810*x^3+915*x^4+615*x^5+228*x^6+36*x^7)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 44*n + 40 + (20*n^2 - 214*n + 194)*x + (64*n^2 - 470*n + 406)*x^2 + (120*n^2 - 582*n + 462)*x^3 + (108*n^2 - 390*n + 282)*x^4 + (36*n^2 - 108*n + 72)*x^5)*P[n - 1, x] - (10*n - 52 + (92*n - 374)*x + (336*n - 1116)*x^2 + (628*n - 1768)*x^3 + (638*n - 1568)*x^4 + (336*n - 738)*x^5 + (72*n - 144)*x^6)*D[P[n - 1, x], x] + (15 + 123*x + 426*x^2 + 810*x^3 + 915*x^4 + 615*x^5 + 228*x^6 + 36*x^7)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 16*n + 8 + (16*n^2 - 32*n + 16)*x + (8*n^2 - 16*n + 8)*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; v[2] = 3 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 44*n + 40 + (20*n^2 - 214*n + 194)*x + (64*n^2 - 470*n + 406)*x^2 + (120*n^2 - 582*n + 462)*x^3 + (108*n^2 - 390*n + 282)*x^4 + (36*n^2 - 108*n + 72)*x^5)*v[i - 1] - (10*n - 52 + (92*n - 374)*x + (336*n - 1116)*x^2 + (628*n - 1768)*x^3 + (638*n - 1568)*x^4 + (336*n - 738)*x^5 + (72*n - 144)*x^6)*deriv(v[i - 1], x) + (15 + 123*x + 426*x^2 + 810*x^3 + 915*x^4 + 615*x^5 + 228*x^6 + 36*x^7)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 16*n + 8 + (16*n^2 - 32*n + 16)*x + (8*n^2 - 16*n + 8)*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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023