Real-rooted atlas

A114656 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 9\,x + 2\,x^{2} - 5\,n x + 2\,n x^{2} - 4\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(2 - 23\,x + 44\,x^{2} - 12\,x^{3} + 37\,n x - 64\,n x^{2} + 12\,n x^{3} - 36\,n^{2} x + 24\,n^{2} x^{2} + 2\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(4\,x - 18\,x^{2} + 24\,x^{3} - 8\,x^{4} - 6\,n x + 20\,n x^{2} - 16\,n x^{3} + 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
3714425
41360488129
52118032014416681
63143014001360400323653
74388246208120504010566419825
8571624125443528039200170242688128108545

All 15 rows

Technical details
Node id
n-49e6adb785ceb927 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-770cb16f18256565

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  3,    2;
   3:  7,   14,     4;
   4: 13,   60,    48,      8;
   5: 21,  180,   320,    144,     16;
   6: 31,  430,  1400,   1360,    400,      32;
   7: 43,  882,  4620,   8120,   5040,    1056,     64;
   8: 57, 1624, 12544,  35280,  39200,   17024,   2688,    128;
   9: 73, 2760, 29568, 122304, 216384,  166656,  53760,   6656,   256;
  10: 91, 4410, 62640, 358848, 931392, 1137024, 645120, 161280, 16128, 512;
%F Formula
The row polynomials satisfy (2*n^3-4*n^2+6+(4*n^2-5*n-9)*x+(2*n+2)*x^2)*P(n,x) = (2*n^3+2+(12*n^3-36*n^2+37*n-23)*x+(24*n^2-64*n+44)*x^2+(12*n-12)*x^3)*P(n-1,x) + ((4*n^2-6*n+4)*x-(8*n^2-20*n+18)*x^2-(16*n-24)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 2 + (12*n^3 - 36*n^2 + 37*n - 23)*x + (24*n^2 - 64*n + 44)*x^2 + (12*n - 12)*x^3)*P[n - 1, x] + ((4*n^2 - 6*n + 4)*x - (8*n^2 - 20*n + 18)*x^2 - (16*n - 24)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 4*n^2 + 6 + (4*n^2 - 5*n - 9)*x + (2*n + 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^3 + 2 + (12*n^3 - 36*n^2 + 37*n - 23)*x + (24*n^2 - 64*n + 44)*x^2 + (12*n - 12)*x^3)*v[i - 1] + ((4*n^2 - 6*n + 4)*x - (8*n^2 - 20*n + 18)*x^2 - (16*n - 24)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 4*n^2 + 6 + (4*n^2 - 5*n - 9)*x + (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, v[i])))};
T(10)

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