Real-rooted atlas

A033282 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 2\,n x + n^{2} - 7\,n^{2} x - 12\,n^{3} x + 2\,n^{3} x^{2} - 4\,n^{4} x + 5\,n^{4} x^{2} + 4\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(16\,x - 28\,x^{2} + 4\,x^{3} + 2\,n + 26\,n x - 98\,n x^{2} + 22\,n x^{3} + n^{2} + n^{2} x - 126\,n^{2} x^{2} + 50\,n^{2} x^{3} - 12\,n^{3} x - 68\,n^{3} x^{2} + 60\,n^{3} x^{3} - 4\,n^{4} x - 9\,n^{4} x^{2} + 40\,n^{4} x^{3} + 4\,n^{5} x^{2} + 14\,n^{5} x^{3} + n^{6} x^{2} + 2\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - 6\,x^{2} - 6\,x^{3} + 2\,x^{4} + 2\,n x - 22\,n x^{2} - 14\,n x^{3} + 10\,n x^{4} - 24\,n^{2} x^{2} - 4\,n^{2} x^{3} + 20\,n^{2} x^{4} - 8\,n^{3} x^{2} + 12\,n^{3} x^{3} + 20\,n^{3} x^{4} + 10\,n^{4} x^{3} + 10\,n^{4} x^{4} + 2\,n^{5} x^{3} + 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111617
2140135176
31725678961536
4111215125376525012251
51160324019200412502851292363
61216607552800185625277992147147669856
7128010395123200625625151351217167157321604721888

All 15 rows

Technical details
Node id
n-cec7a3928f1692d4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-720047eeef937849

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 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: 1,  16;
  2: 1,  40,   135;
  3: 1,  72,   567,    896;
  4: 1, 112,  1512,   5376,    5250;
  5: 1, 160,  3240,  19200,   41250,    28512;
  6: 1, 216,  6075,  52800,  185625,   277992,   147147;
  7: 1, 280, 10395, 123200,  625625,  1513512,  1716715,   732160;
  8: 1, 352, 16632, 256256, 1751750,  6054048, 10986976,  9957376,  3544398;
  9: 1, 432, 25272, 489216, 4299750, 19813248, 50939616, 73322496, 55099278, 16796000;
%F Formula
The row polynomials satisfy (n^2+2*n-(4*n^4+12*n^3+7*n^2-2*n)*x+(n^6+4*n^5+5*n^4+2*n^3)*x^2)*P(n,x) = (n^2+2*n-(4*n^4+12*n^3-n^2-26*n-16)*x+(n^6+4*n^5-9*n^4-68*n^3-126*n^2-98*n-28)*x^2+(2*n^6+14*n^5+40*n^4+60*n^3+50*n^2+22*n+4)*x^3)*P(n-1,x) + ((2*n+2)*x-(8*n^3+24*n^2+22*n+6)*x^2+(2*n^5+10*n^4+12*n^3-4*n^2-14*n-6)*x^3+(2*n^5+10*n^4+20*n^3+20*n^2+10*n+2)*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[((n^2 + 2*n - (4*n^4 + 12*n^3 - n^2 - 26*n - 16)*x + (n^6 + 4*n^5 - 9*n^4 - 68*n^3 - 126*n^2 - 98*n - 28)*x^2 + (2*n^6 + 14*n^5 + 40*n^4 + 60*n^3 + 50*n^2 + 22*n + 4)*x^3)*P[n - 1, x] + ((2*n + 2)*x - (8*n^3 + 24*n^2 + 22*n + 6)*x^2 + (2*n^5 + 10*n^4 + 12*n^3 - 4*n^2 - 14*n - 6)*x^3 + (2*n^5 + 10*n^4 + 20*n^3 + 20*n^2 + 10*n + 2)*x^4)*D[P[n - 1, x], x])/(n^2 + 2*n - (4*n^4 + 12*n^3 + 7*n^2 - 2*n)*x + (n^6 + 4*n^5 + 5*n^4 + 2*n^3)*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] = ((n^2 + 2*n - (4*n^4 + 12*n^3 - n^2 - 26*n - 16)*x + (n^6 + 4*n^5 - 9*n^4 - 68*n^3 - 126*n^2 - 98*n - 28)*x^2 + (2*n^6 + 14*n^5 + 40*n^4 + 60*n^3 + 50*n^2 + 22*n + 4)*x^3)*v[i - 1] + ((2*n + 2)*x - (8*n^3 + 24*n^2 + 22*n + 6)*x^2 + (2*n^5 + 10*n^4 + 12*n^3 - 4*n^2 - 14*n - 6)*x^3 + (2*n^5 + 10*n^4 + 20*n^3 + 20*n^2 + 10*n + 2)*x^4)*deriv(v[i - 1], x))/(n^2 + 2*n - (4*n^4 + 12*n^3 + 7*n^2 - 2*n)*x + (n^6 + 4*n^5 + 5*n^4 + 2*n^3)*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