Real-rooted atlas

A165253 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-300 - 420\,x + 168\,x^{2} + 204\,x^{3} - 12\,x^{4} + 200\,n + 280\,n x - 562\,n x^{2} - 406\,n x^{3} + 44\,n x^{4} + 450\,n^{2} x^{2} + 270\,n^{2} x^{3} - 63\,n^{2} x^{4} - 100\,n^{3} x^{2} - 60\,n^{3} x^{3} + 44\,n^{3} x^{4} - 15\,n^{4} x^{4} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-400 - 610\,x - 216\,x^{2} + 46\,x^{3} + 16\,x^{4} + 300\,n + 295\,n x - 388\,n x^{2} - 423\,n x^{3} - 76\,n x^{4} + 175\,n^{2} x + 710\,n^{2} x^{2} + 510\,n^{2} x^{3} + 68\,n^{2} x^{4} - 4\,n^{2} x^{5} - 150\,n^{3} x^{2} - 115\,n^{3} x^{3} + 11\,n^{3} x^{4} + 8\,n^{3} x^{5} - 10\,n^{4} x^{3} - 22\,n^{4} x^{4} - 5\,n^{4} x^{5} + 3\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(125 + 600\,x + 760\,x^{2} + 244\,x^{3} - 57\,x^{4} - 16\,x^{5} + 350\,n x^{2} + 720\,n x^{3} + 430\,n x^{4} + 60\,n x^{5} - 175\,n^{2} x^{2} - 360\,n^{2} x^{3} - 195\,n^{2} x^{4} - 6\,n^{2} x^{5} + 4\,n^{2} x^{6} - 20\,n^{3} x^{4} - 24\,n^{3} x^{5} - 4\,n^{3} x^{6} + 5\,n^{4} x^{4} + 6\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2246
3520934
413767216177
53426037817625873
68984016471184350364146
723326166453630428756124919142

All 15 rows

Technical details
Node id
n-fdd40bf8469d329f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c91d5d1bb301f35e

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  2:    2,     4;
  3:    5,    20,     9;
  4:   13,    76,    72,     16;
  5:   34,   260,   378,    176,    25;
  6:   89,   840,  1647,   1184,   350,    36;
  7:  233,  2616,  6453,   6304,  2875,   612,    49;
  8:  610,  7940, 23598,  29200, 18125,  5940,   980,   64;
  9: 1597, 23644, 82206, 123248, 97125, 43308, 10976, 1472, 81;
%F Formula
The row polynomials satisfy (200*n-300+(280*n-420)*x-(100*n^3-450*n^2+562*n-168)*x^2-(60*n^3-270*n^2+406*n-204)*x^3+(2*n^5-15*n^4+44*n^3-63*n^2+44*n-12)*x^4)*P(n,x) = (300*n-400+(175*n^2+295*n-610)*x-(150*n^3-710*n^2+388*n+216)*x^2-(10*n^4+115*n^3-510*n^2+423*n-46)*x^3+(3*n^5-22*n^4+11*n^3+68*n^2-76*n+16)*x^4+(n^5-5*n^4+8*n^3-4*n^2)*x^5)*P(n-1,x) + (125+600*x-(175*n^2-350*n-760)*x^2-(360*n^2-720*n-244)*x^3+(5*n^4-20*n^3-195*n^2+430*n-57)*x^4+(6*n^4-24*n^3-6*n^2+60*n-16)*x^5+(n^4-4*n^3+4*n^2)*x^6)*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[((300*n - 400 + (175*n^2 + 295*n - 610)*x - (150*n^3 - 710*n^2 + 388*n + 216)*x^2 - (10*n^4 + 115*n^3 - 510*n^2 + 423*n - 46)*x^3 + (3*n^5 - 22*n^4 + 11*n^3 + 68*n^2 - 76*n + 16)*x^4 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^5)*P[n - 1, x] + (125 + 600*x - (175*n^2 - 350*n - 760)*x^2 - (360*n^2 - 720*n - 244)*x^3 + (5*n^4 - 20*n^3 - 195*n^2 + 430*n - 57)*x^4 + (6*n^4 - 24*n^3 - 6*n^2 + 60*n - 16)*x^5 + (n^4 - 4*n^3 + 4*n^2)*x^6)*D[P[n - 1, x], x])/(200*n - 300 + (280*n - 420)*x - (100*n^3 - 450*n^2 + 562*n - 168)*x^2 - (60*n^3 - 270*n^2 + 406*n - 204)*x^3 + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*x^4)]];
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] = ((300*n - 400 + (175*n^2 + 295*n - 610)*x - (150*n^3 - 710*n^2 + 388*n + 216)*x^2 - (10*n^4 + 115*n^3 - 510*n^2 + 423*n - 46)*x^3 + (3*n^5 - 22*n^4 + 11*n^3 + 68*n^2 - 76*n + 16)*x^4 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^5)*v[i - 1] + (125 + 600*x - (175*n^2 - 350*n - 760)*x^2 - (360*n^2 - 720*n - 244)*x^3 + (5*n^4 - 20*n^3 - 195*n^2 + 430*n - 57)*x^4 + (6*n^4 - 24*n^3 - 6*n^2 + 60*n - 16)*x^5 + (n^4 - 4*n^3 + 4*n^2)*x^6)*deriv(v[i - 1], x))/(200*n - 300 + (280*n - 420)*x - (100*n^3 - 450*n^2 + 562*n - 168)*x^2 - (60*n^3 - 270*n^2 + 406*n - 204)*x^3 + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*x^4)); 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