Real-rooted atlas

A108838 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 6\,x + 3\,x^{2} + 4\,n - 2\,n x + 4\,n x^{2} + 4\,n^{2} + 6\,n^{2} x + n^{2} x^{2} + 4\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(6 - 3\,x + 3\,x^{3} + 14\,n + 2\,n x + 4\,n x^{2} + 6\,n x^{3} + 13\,n^{2} + 12\,n^{2} x + 10\,n^{2} x^{2} + 3\,n^{2} x^{3} + 6\,n^{3} + 10\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + 3\,x^{3} - 3\,x^{4} + 6\,n x - 4\,n x^{3} - 2\,n x^{4} + 6\,n^{2} x - 2\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1538
21016430
31755355112
426144180646420
53731566545510571584
6506081960224098016086006
76510714914852661741890231922880

All 15 rows

Technical details
Node id
n-f1f267c0a4c340f1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-20d00a65c074974f

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:   2;
  1:   5,    3;
  2:  10,   16,     4;
  3:  17,   55,    35,     5;
  4:  26,  144,   180,    64,      6;
  5:  37,  315,   665,   455,    105,     7;
  6:  50,  608,  1960,  2240,    980,   160,     8;
  7:  65, 1071,  4914,  8526,   6174,  1890,   231,    9;
  8:  82, 1760, 10920, 26880,  29400, 14784,  3360,  320,  10;
  9: 101, 2739, 22110, 73458, 113652, 85932, 31878, 5610, 429, 11;
%F Formula
The row polynomials satisfy (n^4+4*n^3+4*n^2+4*n+3+(2*n^3+6*n^2-2*n-6)*x+(n^2+4*n+3)*x^2)*P(n,x) = (n^4+6*n^3+13*n^2+14*n+6+(3*n^4+10*n^3+12*n^2+2*n-3)*x+(6*n^3+10*n^2+4*n)*x^2+(3*n^2+6*n+3)*x^3)*P(n-1,x) + ((2*n^3+6*n^2+6*n+3)*x-(2*n^3+2*n^2+3)*x^2-(4*n^2+4*n-3)*x^3-(2*n+3)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 6*n^3 + 13*n^2 + 14*n + 6 + (3*n^4 + 10*n^3 + 12*n^2 + 2*n - 3)*x + (6*n^3 + 10*n^2 + 4*n)*x^2 + (3*n^2 + 6*n + 3)*x^3)*P[n - 1, x] + ((2*n^3 + 6*n^2 + 6*n + 3)*x - (2*n^3 + 2*n^2 + 3)*x^2 - (4*n^2 + 4*n - 3)*x^3 - (2*n + 3)*x^4)*D[P[n - 1, x], x])/(n^4 + 4*n^3 + 4*n^2 + 4*n + 3 + (2*n^3 + 6*n^2 - 2*n - 6)*x + (n^2 + 4*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] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((n^4 + 6*n^3 + 13*n^2 + 14*n + 6 + (3*n^4 + 10*n^3 + 12*n^2 + 2*n - 3)*x + (6*n^3 + 10*n^2 + 4*n)*x^2 + (3*n^2 + 6*n + 3)*x^3)*v[i - 1] + ((2*n^3 + 6*n^2 + 6*n + 3)*x - (2*n^3 + 2*n^2 + 3)*x^2 - (4*n^2 + 4*n - 3)*x^3 - (2*n + 3)*x^4)*deriv(v[i - 1], x))/(n^4 + 4*n^3 + 4*n^2 + 4*n + 3 + (2*n^3 + 6*n^2 - 2*n - 6)*x + (n^2 + 4*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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023