Real-rooted atlas

A103327 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 5\,x - 2\,x^{2} - n - 14\,n x + 4\,n x^{2} - 4\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(12 + 26\,x + 7\,x^{2} - 10\,x^{3} - 28\,n - 67\,n x - 22\,n x^{2} + 12\,n x^{3} + 24\,n^{2} x + 24\,n^{2} x^{2} + 16\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x + 10\,x^{2} - 4\,x^{3} - 8\,x^{4} - 8\,n x - 24\,n x^{2} - 16\,n x^{3} - 8\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1516
22814143
3144128271300
47049283604412037
533285888347280065113554
61536034304276489856154090188799
769632188416194304960002352026881191574680

All 15 rows

Technical details
Node id
n-9ec8adace7cff58f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0e8a7595806f7ab3

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:      28,      14,       1;
  3:     144,     128,      27,       1;
  4:     704,     928,     360,      44,       1;
  5:    3328,    5888,    3472,     800,      65,      1;
  6:   15360,   34304,   27648,    9856,    1540,     90,     1;
  7:   69632,  188416,  194304,   96000,   23520,   2688,   119,    1;
  8:  311296,  991232, 1251328,  799744,  274560,  49728,  4368,  152,   1;
  9: 1376256, 5046272, 7557120, 5963776, 2672384, 684288, 96096, 6720, 189, 1;
%F Formula
The row polynomials satisfy (4*n^3-4*n^2-n+1+(8*n^2-14*n+5)*x+(4*n-2)*x^2)*P(n,x) = (16*n^3-28*n+12+(12*n^3+24*n^2-67*n+26)*x+(24*n^2-22*n+7)*x^2+(12*n-10)*x^3)*P(n-1,x) - ((8*n^2+8*n-6)*x+(8*n^2+24*n-10)*x^2+(16*n+4)*x^3+8*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[((16*n^3 - 28*n + 12 + (12*n^3 + 24*n^2 - 67*n + 26)*x + (24*n^2 - 22*n + 7)*x^2 + (12*n - 10)*x^3)*P[n - 1, x] - ((8*n^2 + 8*n - 6)*x + (8*n^2 + 24*n - 10)*x^2 + (16*n + 4)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 4*n^2 - n + 1 + (8*n^2 - 14*n + 5)*x + (4*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] = ((16*n^3 - 28*n + 12 + (12*n^3 + 24*n^2 - 67*n + 26)*x + (24*n^2 - 22*n + 7)*x^2 + (12*n - 10)*x^3)*v[i - 1] - ((8*n^2 + 8*n - 6)*x + (8*n^2 + 24*n - 10)*x^2 + (16*n + 4)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 4*n^2 - n + 1 + (8*n^2 - 14*n + 5)*x + (4*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. [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