Real-rooted atlas

A110441 → basis A007318 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(6\,n - 10\,n x - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(8 - 24\,x - 10\,x^{2} - x^{3} + 16\,n - 22\,n x - 10\,n x^{2} - n x^{3} + 12\,n^{2} + 3\,n^{2} x + 4\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-32 - 88\,x - 68\,x^{2} - 15\,x^{3} - x^{4} + 8\,n + 8\,n x + n x^{2} + 8\,n^{2} + 8\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1516
22210133
39268151176
4372388138201919
51468199610002322514722
6568895846264204035030123957
7217284380835544150803620492351120308

All 15 rows

Technical details
Node id
n-bbb26555c9990c61 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0a3f99df0c7231cc

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:      5,      1;
  2:     22,     10,      1;
  3:     92,     68,     15,      1;
  4:    372,    388,    138,     20,      1;
  5:   1468,   1996,   1000,    232,     25,     1;
  6:   5688,   9584,   6264,   2040,    350,    30,    1;
  7:  21728,  43808,  35544,  15080,   3620,   492,   35,   1;
  8:  82064, 192992, 187760,  98864,  30860,  5852,  658,  40,  1;
  9: 307088, 826192, 939552, 593440, 229684, 56532, 8848, 848, 45, 1;
%F Formula
The row polynomials satisfy (2*n^3+6*n-10*n*x-2*n*x^2)*P(n,x) = (4*n^3+12*n^2+16*n+8+(n^3+3*n^2-22*n-24)*x-(10*n+10)*x^2-(n+1)*x^3)*P(n-1,x) + (8*n^2+8*n-32+(8*n^2+8*n-88)*x+(n^2+n-68)*x^2-15*x^3-x^4)*d/dx 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[((4*n^3 + 12*n^2 + 16*n + 8 + (n^3 + 3*n^2 - 22*n - 24)*x - (10*n + 10)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (8*n^2 + 8*n - 32 + (8*n^2 + 8*n - 88)*x + (n^2 + n - 68)*x^2 - 15*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 + 6*n - 10*n*x - 2*n*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] = ((4*n^3 + 12*n^2 + 16*n + 8 + (n^3 + 3*n^2 - 22*n - 24)*x - (10*n + 10)*x^2 - (n + 1)*x^3)*v[i - 1] + (8*n^2 + 8*n - 32 + (8*n^2 + 8*n - 88)*x + (n^2 + n - 68)*x^2 - 15*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 + 6*n - 10*n*x - 2*n*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023