Real-rooted atlas

A078812 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x - 2\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(10 + 52\,x + 98\,x^{2} + 80\,x^{3} + 24\,x^{4} - 2\,n - 16\,n x - 38\,n x^{2} - 36\,n x^{3} - 12\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 31\,x + 75\,x^{2} + 89\,x^{3} + 52\,x^{4} + 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
38221545
4218812256287
5553106516042091829
6144102028744028280978011655
7377321611376213642247112554291174269
8987984841934988001391301171205458010864473263

All 15 rows

Technical details
Node id
n-872dde5ceb16245b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-63c829d69f89a5ff

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    3,     4;
   3:    8,    22,     15;
   4:   21,    88,    122,      56;
   5:   55,   310,    651,     604,     209;
   6:  144,  1020,   2874,    4028,    2809,     780;
   7:  377,  3216,  11376,   21364,   22471,   12554,    2911;
   8:  987,  9848,  41934,   98800,  139130,  117120,   54580,   10864;
   9: 2584, 29522, 147009,  416804,  736015,  829020,  581730,  232536,  40545;
  10: 6765, 87080, 496498, 1645932, 3496597, 4936940, 4633320, 2787384, 975471, 151316;
%F Formula
The row polynomials satisfy (2*n-2+(4*n-4)*x+(2*n-2)*x^2)*P(n,x) = -(2*n-10+(16*n-52)*x+(38*n-98)*x^2+(36*n-80)*x^3+(12*n-24)*x^4)*P(n-1,x) + (5+31*x+75*x^2+89*x^3+52*x^4+12*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 10 + (16*n - 52)*x + (38*n - 98)*x^2 + (36*n - 80)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*D[P[n - 1, x], x])/(2*n - 2 + (4*n - 4)*x + (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 10 + (16*n - 52)*x + (38*n - 98)*x^2 + (36*n - 80)*x^3 + (12*n - 24)*x^4)*v[i - 1] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*deriv(v[i - 1], x))/(2*n - 2 + (4*n - 4)*x + (2*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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