Real-rooted atlas

A229032 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n + 11\,n x + 8\,n^{2} x + 2\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-36 - 105\,x - 88\,x^{2} - 16\,x^{3} + 60\,n + 91\,n x + 18\,n x^{2} - 8\,n x^{3} + 52\,n^{2} x + 68\,n^{2} x^{2} + 16\,n^{2} x^{3} + 10\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-36 - 150\,x - 226\,x^{2} - 144\,x^{3} - 32\,x^{4} - 48\,n x - 148\,n x^{2} - 148\,n x^{3} - 48\,n x^{4} - 12\,n^{2} x^{2} - 28\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
122
27815
3203252
46114448253
51825442881014
6547200815842564395
7164071047296204818088

All 15 rows

Technical details
Node id
n-389a349557234c94 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-784e16ba1d83baa5

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • 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:     2;
  2:     7,     8;
  3:    20,    32;
  4:    61,   144,     48;
  5:   182,   544,    288;
  6:   547,  2008,   1584,   256;
  7:  1640,  7104,   7296,  2048;
  8:  4921, 24608,  31392, 13312,  1280;
  9: 14762, 83648, 127296, 71680, 12800;
%F Formula
The row polynomials satisfy (12*n+(8*n^2+11*n)*x+(2*n^3+2*n^2)*x^2)*P(n,x) = (60*n-36+(52*n^2+91*n-105)*x+(10*n^3+68*n^2+18*n-88)*x^2+(8*n^3+16*n^2-8*n-16)*x^3)*P(n-1,x) - (36+(48*n+150)*x+(12*n^2+148*n+226)*x^2+(28*n^2+148*n+144)*x^3+(16*n^2+48*n+32)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((60*n - 36 + (52*n^2 + 91*n - 105)*x + (10*n^3 + 68*n^2 + 18*n - 88)*x^2 + (8*n^3 + 16*n^2 - 8*n - 16)*x^3)*P[n - 1, x] - (36 + (48*n + 150)*x + (12*n^2 + 148*n + 226)*x^2 + (28*n^2 + 148*n + 144)*x^3 + (16*n^2 + 48*n + 32)*x^4)*D[P[n - 1, x], x])/(12*n + (8*n^2 + 11*n)*x + (2*n^3 + 2*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((60*n - 36 + (52*n^2 + 91*n - 105)*x + (10*n^3 + 68*n^2 + 18*n - 88)*x^2 + (8*n^3 + 16*n^2 - 8*n - 16)*x^3)*v[i - 1] - (36 + (48*n + 150)*x + (12*n^2 + 148*n + 226)*x^2 + (28*n^2 + 148*n + 144)*x^3 + (16*n^2 + 48*n + 32)*x^4)*deriv(v[i - 1], x))/(12*n + (8*n^2 + 11*n)*x + (2*n^3 + 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 - 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