Real-rooted atlas

A114656 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n + 6\,n x - n x^{2} + 7\,n^{2} - 8\,n^{2} x + n^{2} x^{2} - 4\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(-10\,x^{2} + 2\,x^{3} - 4\,n + 6\,n x + 27\,n x^{2} - 5\,n x^{3} + 6\,n^{2} - n^{2} x - 23\,n^{2} x^{2} + 3\,n^{2} x^{3} - 4\,n^{3} - 8\,n^{3} x + 6\,n^{3} x^{2} + 2\,n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(8\,x^{2} - 6\,x^{3} + x^{4} + 4\,n x - 22\,n x^{2} + 14\,n x^{3} - 2\,n x^{4} - 8\,n^{2} x + 12\,n^{2} x^{2} - 4\,n^{2} x^{3} + 4\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
31618337
456120484228
517664048010051401
651228803600140018068578
714081142421840140003360294752333
83712412161128961097604312070564488318216

All 15 rows

Technical details
Node id
n-1d21d78d875f8903 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a8b715cdd5f5d5e8

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:     4,      2;
   3:    16,     18,       3;
   4:    56,    120,      48,       4;
   5:   176,    640,     480,     100,       5;
   6:   512,   2880,    3600,    1400,     180,       6;
   7:  1408,  11424,   21840,   14000,    3360,     294,      7;
   8:  3712,  41216,  112896,  109760,   43120,    7056,    448,     8;
   9:  9472, 138240,  516096,  715008,  423360,  112896,  13440,   648,   9;
  10: 23552, 437760, 2142720, 4032000, 3386880, 1354752, 262080, 23760, 900, 10;
%F Formula
The row polynomials satisfy (n^4-4*n^3+7*n^2-4*n+(2*n^3-8*n^2+6*n)*x+(n^2-n)*x^2)*P(n,x) = (2*n^4-4*n^3+6*n^2-4*n+(3*n^4-8*n^3-n^2+6*n)*x+(6*n^3-23*n^2+27*n-10)*x^2+(3*n^2-5*n+2)*x^3)*P(n-1,x) + ((4*n^3-8*n^2+4*n)*x-(2*n^3-12*n^2+22*n-8)*x^2-(4*n^2-14*n+6)*x^3-(2*n-1)*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^4 - 4*n^3 + 6*n^2 - 4*n + (3*n^4 - 8*n^3 - n^2 + 6*n)*x + (6*n^3 - 23*n^2 + 27*n - 10)*x^2 + (3*n^2 - 5*n + 2)*x^3)*P[n - 1, x] + ((4*n^3 - 8*n^2 + 4*n)*x - (2*n^3 - 12*n^2 + 22*n - 8)*x^2 - (4*n^2 - 14*n + 6)*x^3 - (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^3 + 7*n^2 - 4*n + (2*n^3 - 8*n^2 + 6*n)*x + (n^2 - n)*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^4 - 4*n^3 + 6*n^2 - 4*n + (3*n^4 - 8*n^3 - n^2 + 6*n)*x + (6*n^3 - 23*n^2 + 27*n - 10)*x^2 + (3*n^2 - 5*n + 2)*x^3)*v[i - 1] + ((4*n^3 - 8*n^2 + 4*n)*x - (2*n^3 - 12*n^2 + 22*n - 8)*x^2 - (4*n^2 - 14*n + 6)*x^3 - (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^3 + 7*n^2 - 4*n + (2*n^3 - 8*n^2 + 6*n)*x + (n^2 - n)*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