Real-rooted atlas

A207606 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-90\,x - 66\,x^{2} + 12\,x^{3} - 20\,n - 66\,n x - 48\,n x^{2} + 26\,n x^{3} + 18\,n^{2} x^{3} + 2\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n}(x) = \left(20 - 134\,x - 266\,x^{2} - 64\,x^{3} + 12\,x^{4} - 30\,n - 119\,n x - 183\,n x^{2} - 2\,n x^{3} + 22\,n x^{4} - 5\,n^{2} x - 12\,n^{2} x^{2} + 29\,n^{2} x^{3} + 12\,n^{2} x^{4} + 3\,n^{3} x^{2} + 7\,n^{3} x^{3} + 2\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-25 - 145\,x - 243\,x^{2} - 113\,x^{3} + 16\,x^{4} + 6\,x^{5} + 5\,n x^{2} + 46\,n x^{3} + 49\,n x^{4} + 8\,n x^{5} + 5\,n^{2} x^{2} + 16\,n^{2} x^{3} + 13\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 13 + 22\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01322641
13488518181
2893102799210780
323310201251640145123301
461032165004349612252101413775
5159798481858516520790020882871656841
641812952265523708444312515534328337618232406
710946870802222672833762100309626427685486447720943009

All 15 rows

Technical details
Node id
n-322cc71514bc6b58 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-270b78919d21a4f3

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:    13,     22,       6;
  1:    34,     88,      51,       8;
  2:    89,    310,     279,      92,      10;
  3:   233,   1020,    1251,     640,     145,      12;
  4:   610,   3216,    5004,    3496,    1225,     210,      14;
  5:  1597,   9848,   18585,   16520,    7900,    2088,     287,     16;
  6:  4181,  29522,   65523,   70844,   43125,   15534,    3283,    376,    18;
  7: 10946,  87080,  222267,  283376,  210030,   96264,   27685,   4864,   477,   20;
  8: 28657, 253610,  731898, 1075640,  941185,  525294,  192080,  45856,  6885,  590,  22;
  9: 75025, 731064, 2353842, 3919408, 3958075, 2609040, 1161398, 352224, 71766, 9400, 715, 24;
%F Formula
The row polynomials satisfy -(20*n+(66*n+90)*x-(2*n^3-48*n-66)*x^2-(4*n^3+18*n^2+26*n+12)*x^3)*P(n,x) = -(30*n-20+(5*n^2+119*n+134)*x-(3*n^3-12*n^2-183*n-266)*x^2-(7*n^3+29*n^2-2*n-64)*x^3-(2*n^3+12*n^2+22*n+12)*x^4)*P(n-1,x) - (25+145*x-(5*n^2+5*n-243)*x^2-(16*n^2+46*n-113)*x^3-(13*n^2+49*n+16)*x^4-(2*n^2+8*n+6)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 22*x + 6*x^2.
%t Mathematica
P[0, x_] := 13 + 22*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 20 + (5*n^2 + 119*n + 134)*x - (3*n^3 - 12*n^2 - 183*n - 266)*x^2 - (7*n^3 + 29*n^2 - 2*n - 64)*x^3 - (2*n^3 + 12*n^2 + 22*n + 12)*x^4)*P[n - 1, x] - (25 + 145*x - (5*n^2 + 5*n - 243)*x^2 - (16*n^2 + 46*n - 113)*x^3 - (13*n^2 + 49*n + 16)*x^4 - (2*n^2 + 8*n + 6)*x^5)*D[P[n - 1, x], x])/(-20*n - (66*n + 90)*x + (2*n^3 - 48*n - 66)*x^2 + (4*n^3 + 18*n^2 + 26*n + 12)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 13 + 22*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(30*n - 20 + (5*n^2 + 119*n + 134)*x - (3*n^3 - 12*n^2 - 183*n - 266)*x^2 - (7*n^3 + 29*n^2 - 2*n - 64)*x^3 - (2*n^3 + 12*n^2 + 22*n + 12)*x^4)*v[i - 1] - (25 + 145*x - (5*n^2 + 5*n - 243)*x^2 - (16*n^2 + 46*n - 113)*x^3 - (13*n^2 + 49*n + 16)*x^4 - (2*n^2 + 8*n + 6)*x^5)*deriv(v[i - 1], x))/(-20*n - (66*n + 90)*x + (2*n^3 - 48*n - 66)*x^2 + (4*n^3 + 18*n^2 + 26*n + 12)*x^3)); 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