Real-rooted atlas

A123519 → 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

\(\left(3 + 6\,x + 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(1 - 7\,x - 33\,x^{2} - 41\,x^{3} - 16\,x^{4} - 2\,n - 18\,n x - 46\,n x^{2} - 46\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}(x) + \left(6 + 38\,x + 94\,x^{2} + 114\,x^{3} + 68\,x^{4} + 16\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 11 + 36\,x + 29\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
011362976
141199319169728
2153984235824969856976
35714573145782312618259574166848
42131204288127117177620345712805233461640576
57953887834233571117875176539516675658724151950256138368
629681378160210186866568961314055016555888130020285819616113668958821376
7110771158603710067556371854208807907413875062214537050097683212382019876625109563660288

All 15 rows

Technical details
Node id
n-e142315f0fa5c93e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-83ba96b3decd27c9

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:      11,       36,        29;
  1:      41,      199,       319,        169;
  2:     153,      984,      2358,       2496,        985;
  3:     571,     4573,     14578,      23126,      18259,       5741;
  4:    2131,    20428,     81271,     171776,     203457,     128052,       33461;
  5:    7953,    88783,    423357,    1117875,    1765395,    1667565,      872415,      195025;
  6:   29681,   378160,   2101868,    6656896,   13140550,   16555888,    13002028,     5819616,    1136689;
  7:  110771,  1586037,  10067556,   37185420,   88079074,  138750622,   145370500,    97683212,   38201987,    6625109;
  8:  413403,  6571252,  46898577,  197908224,  546872606, 1033983048,  1354731378,  1214576288,  713129271,  247619364,   38613965;
  9: 1542841, 26957847, 213671419, 1014117789, 3202422298, 7065073638, 11111872982, 12459518586, 9761035901, 5088540787, 1588719839, 225058681;
%F Formula
The row polynomials satisfy (2*n+3+(4*n+6)*x+(2*n+3)*x^2)*P(n,x) = -(2*n-1+(18*n+7)*x+(46*n+33)*x^2+(46*n+41)*x^3+(16*n+16)*x^4)*P(n-1,x) + (6+38*x+94*x^2+114*x^3+68*x^4+16*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 36*x + 29*x^2.
%t Mathematica
P[0, x_] := 11 + 36*x + 29*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + (18*n + 7)*x + (46*n + 33)*x^2 + (46*n + 41)*x^3 + (16*n + 16)*x^4)*P[n - 1, x] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*D[P[n - 1, x], x])/(2*n + 3 + (4*n + 6)*x + (2*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 11 + 36*x + 29*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 1 + (18*n + 7)*x + (46*n + 33)*x^2 + (46*n + 41)*x^3 + (16*n + 16)*x^4)*v[i - 1] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*deriv(v[i - 1], x))/(2*n + 3 + (4*n + 6)*x + (2*n + 3)*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