Real-rooted atlas

A126216 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 6\,x - 2\,x^{2} + 3\,n - 7\,n x + n x^{2} - 2\,n^{2} - n^{2} x + n^{2} x^{2} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(4\,x^{2} - 2\,x^{3} - 2\,n - 8\,n x - n x^{3} + 2\,n^{2} - 7\,n^{2} x - 4\,n^{2} x^{2} + 3\,n^{2} x^{3} + 2\,n^{3} + 6\,n^{3} x + 6\,n^{3} x^{2} + 4\,n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(x^{2} - x^{4} - 2\,n x - 2\,n x^{2} - 2\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x - 6\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2729
33924366
4196196544450
5930132060010052955
64257792051751440165618963
71901944044381151540029752527119812

All 15 rows

Technical details
Node id
n-3b1f4c325903238e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-51d076129291b50f

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      1;
  2:      7,       2;
  3:     39,      24,       3;
  4:    196,     196,      54,       4;
  5:    930,    1320,     600,     100,      5;
  6:   4257,    7920,    5175,    1440,    165,     6;
  7:  19019,   44044,   38115,   15400,   2975,   252,    7;
  8:  83512,  232232,  252252,  136136,  38500,  5544,  364,   8;
  9: 361998, 1177488, 1546272, 1054872, 401310, 85176, 9576, 504, 9;
%F Formula
The row polynomials satisfy (n^4-2*n^2+3*n-2+(2*n^3-n^2-7*n+6)*x+(n^2+n-2)*x^2)*P(n,x) = (4*n^4+2*n^3+2*n^2-2*n+(3*n^4+6*n^3-7*n^2-8*n)*x+(6*n^3-4*n^2+4)*x^2+(3*n^2-n-2)*x^3)*P(n-1,x) - ((2*n^3+2*n^2+2*n)*x+(2*n^3+6*n^2+2*n-1)*x^2+(4*n^2+2*n)*x^3+(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^4 + 2*n^3 + 2*n^2 - 2*n + (3*n^4 + 6*n^3 - 7*n^2 - 8*n)*x + (6*n^3 - 4*n^2 + 4)*x^2 + (3*n^2 - n - 2)*x^3)*P[n - 1, x] - ((2*n^3 + 2*n^2 + 2*n)*x + (2*n^3 + 6*n^2 + 2*n - 1)*x^2 + (4*n^2 + 2*n)*x^3 + (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 2*n^2 + 3*n - 2 + (2*n^3 - n^2 - 7*n + 6)*x + (n^2 + 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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^4 + 2*n^3 + 2*n^2 - 2*n + (3*n^4 + 6*n^3 - 7*n^2 - 8*n)*x + (6*n^3 - 4*n^2 + 4)*x^2 + (3*n^2 - n - 2)*x^3)*v[i - 1] - ((2*n^3 + 2*n^2 + 2*n)*x + (2*n^3 + 6*n^2 + 2*n - 1)*x^2 + (4*n^2 + 2*n)*x^3 + (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 2*n^2 + 3*n - 2 + (2*n^3 - n^2 - 7*n + 6)*x + (n^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 - 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