Real-rooted atlas

A391045 → θ+1 → D

nonnegative coefficients

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

Recurrence

\(\left(24 - 64\,n + 62\,n^{2} - 26\,n^{3} + 4\,n^{4}\right)\,P_{n}(x) = \left(6 + 396\,x - 22\,n - 900\,n x + 30\,n^{2} + 752\,n^{2} x - 18\,n^{3} - 272\,n^{3} x + 4\,n^{4} + 36\,n^{4} x\right)\,P_{n-1}(x) + \left(-38\,x + 292\,x^{2} + 70\,n x - 516\,n x^{2} - 42\,n^{2} x + 294\,n^{2} x^{2} + 8\,n^{3} x - 52\,n^{3} x^{2}\right)\,P_{n-1}'(x) + \left(-10\,x^{2} + 40\,x^{3} + 14\,n x^{2} - 56\,n x^{3} - 5\,n^{2} x^{2} + 20\,n^{2} x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
222
38614
418601290
53225224020544
6507201680700303180
77216507200756016804218204
8983276231004620026460352856102718

All 15 rows

Technical details
Node id
n-f49329f1f953477f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d23147dfbcf2dc56

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 -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   0;
   2:   2;
   3:   8,    6;
   4:  18,   60,     12;
   5:  32,  252,    240,     20;
   6:  50,  720,   1680,    700,      30;
   7:  72, 1650,   7200,   7560,    1680,     42;
   8:  98, 3276,  23100,  46200,   26460,   3528,     56;
   9: 128, 5880,  61152, 200200,  221760,  77616,   6720,    72;
  10: 162, 9792, 141120, 687960, 1261260, 864864, 199584, 11880, 90;
%F Formula
The row polynomials satisfy (4*n^4-26*n^3+62*n^2-64*n+24)*P(n,x) = (4*n^4-18*n^3+30*n^2-22*n+6+(36*n^4-272*n^3+752*n^2-900*n+396)*x)*P(n-1,x) + ((8*n^3-42*n^2+70*n-38)*x-(52*n^3-294*n^2+516*n-292)*x^2)*d/dx P(n-1,x) - ((5*n^2-14*n+10)*x^2-(20*n^2-56*n+40)*x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^4 - 18*n^3 + 30*n^2 - 22*n + 6 + (36*n^4 - 272*n^3 + 752*n^2 - 900*n + 396)*x)*P[n - 1, x] + ((8*n^3 - 42*n^2 + 70*n - 38)*x - (52*n^3 - 294*n^2 + 516*n - 292)*x^2)*D[P[n - 1, x], x] - ((5*n^2 - 14*n + 10)*x^2 - (20*n^2 - 56*n + 40)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^4 - 26*n^3 + 62*n^2 - 64*n + 24)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((4*n^4 - 18*n^3 + 30*n^2 - 22*n + 6 + (36*n^4 - 272*n^3 + 752*n^2 - 900*n + 396)*x)*v[i - 1] + ((8*n^3 - 42*n^2 + 70*n - 38)*x - (52*n^3 - 294*n^2 + 516*n - 292)*x^2)*deriv(v[i - 1], x) - ((5*n^2 - 14*n + 10)*x^2 - (20*n^2 - 56*n + 40)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^4 - 26*n^3 + 62*n^2 - 64*n + 24)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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