Real-rooted atlas

A367270 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n - 3\,n^{2} + n^{2} x + 2\,n^{3} - 2\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(-2 - 4\,x - 2\,x^{2} + 10\,n + 16\,n x + 5\,n x^{2} - 16\,n^{2} - 18\,n^{2} x - 2\,n^{2} x^{2} + 8\,n^{3} + 2\,n^{3} x - 4\,n^{3} x^{2} + 4\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-x - 2\,x^{2} - x^{3} + 4\,n x + 6\,n x^{2} + 2\,n x^{3} - 4\,n^{2} x - 4\,n^{2} x^{2} - 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2347
31024943
4351209016261
5126560630240251581
6462252037802240525369563
71716110882079016800630010084957751

All 15 rows

Technical details
Node id
n-a10502e10fb179e0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-555bf30a4e16df31

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:     1;
  2:     3,      4;
  3:    10,     24,      9;
  4:    35,    120,     90,     16;
  5:   126,    560,    630,    240,     25;
  6:   462,   2520,   3780,   2240,    525,     36;
  7:  1716,  11088,  20790,  16800,   6300,   1008,    49;
  8:  6435,  48048, 108108, 110880,  57750,  15120,  1764,   64;
  9: 24310, 205920, 540540, 672672, 450450, 166320, 32340, 2880, 81;
%F Formula
The row polynomials satisfy (2*n^3-3*n^2+n+(n^4-2*n^3+n^2)*x)*P(n,x) = (8*n^3-16*n^2+10*n-2+(4*n^4+2*n^3-18*n^2+16*n-4)*x+(3*n^4-4*n^3-2*n^2+5*n-2)*x^2)*P(n-1,x) - ((4*n^2-4*n+1)*x+(2*n^3+4*n^2-6*n+2)*x^2+(2*n^3-2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 16*n^2 + 10*n - 2 + (4*n^4 + 2*n^3 - 18*n^2 + 16*n - 4)*x + (3*n^4 - 4*n^3 - 2*n^2 + 5*n - 2)*x^2)*P[n - 1, x] - ((4*n^2 - 4*n + 1)*x + (2*n^3 + 4*n^2 - 6*n + 2)*x^2 + (2*n^3 - 2*n + 1)*x^3)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + n + (n^4 - 2*n^3 + n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^3 - 16*n^2 + 10*n - 2 + (4*n^4 + 2*n^3 - 18*n^2 + 16*n - 4)*x + (3*n^4 - 4*n^3 - 2*n^2 + 5*n - 2)*x^2)*v[i - 1] - ((4*n^2 - 4*n + 1)*x + (2*n^3 + 4*n^2 - 6*n + 2)*x^2 + (2*n^3 - 2*n + 1)*x^3)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + n + (n^4 - 2*n^3 + n^2)*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

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