Real-rooted atlas

A108838 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 + 4\,n + 3\,n x + 2\,n^{2} - 5\,n^{2} x + n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(12 - 16\,x + 4\,x^{2} + 10\,n - 2\,n x - 4\,n x^{2} + 2\,n^{2} + 13\,n^{2} x - 5\,n^{2} x^{2} + 4\,n^{3} x + 2\,n^{3} x^{2} + n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(4\,x - 6\,x^{2} + 2\,x^{3} + 4\,n x - 3\,n x^{2} - n x^{3} + 3\,n^{2} x^{2} - 3\,n^{2} x^{3} + 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
28816
320601898
44024021632528
5707001260560502640
611216805040448012007212584
7168352815876235201260022689858058

All 15 rows

Technical details
Node id
n-f62ca335c3d5c0c9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b7476c2fe04f345b

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:   2;
  2:   8,     8;
  3:  20,    60,    18;
  4:  40,   240,   216,     32;
  5:  70,   700,  1260,    560,     50;
  6: 112,  1680,  5040,   4480,   1200,     72;
  7: 168,  3528, 15876,  23520,  12600,   2268,    98;
  8: 240,  6720, 42336,  94080,  84000,  30240,  3920,  128;
  9: 330, 11880, 99792, 310464, 415800, 249480, 64680, 6336, 162;
%F Formula
The row polynomials satisfy (2*n^2+4*n-6+(n^4+n^3-5*n^2+3*n)*x)*P(n,x) = (2*n^2+10*n+12+(n^4+4*n^3+13*n^2-2*n-16)*x+(3*n^4+2*n^3-5*n^2-4*n+4)*x^2)*P(n-1,x) + ((4*n+4)*x+(2*n^3+3*n^2-3*n-6)*x^2-(2*n^3+3*n^2+n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 10*n + 12 + (n^4 + 4*n^3 + 13*n^2 - 2*n - 16)*x + (3*n^4 + 2*n^3 - 5*n^2 - 4*n + 4)*x^2)*P[n - 1, x] + ((4*n + 4)*x + (2*n^3 + 3*n^2 - 3*n - 6)*x^2 - (2*n^3 + 3*n^2 + n - 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 + 4*n - 6 + (n^4 + n^3 - 5*n^2 + 3*n)*x)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 10*n + 12 + (n^4 + 4*n^3 + 13*n^2 - 2*n - 16)*x + (3*n^4 + 2*n^3 - 5*n^2 - 4*n + 4)*x^2)*v[i - 1] + ((4*n + 4)*x + (2*n^3 + 3*n^2 - 3*n - 6)*x^2 - (2*n^3 + 3*n^2 + n - 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 + 4*n - 6 + (n^4 + n^3 - 5*n^2 + 3*n)*x)); 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. [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