Real-rooted atlas

A094816 → D → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(120 - 74\,n + 15\,n^{2} - n^{3}\right)\,P_{n-6}(x) + \left(-240 + 60\,x + 168\,n - 27\,n x - 39\,n^{2} + 3\,n^{2} x + 3\,n^{3}\right)\,P_{n-5}(x) + \left(100 - 84\,x + 12\,x^{2} - 97\,n + 45\,n x - 3\,n x^{2} + 30\,n^{2} - 6\,n^{2} x - 3\,n^{3}\right)\,P_{n-4}(x) + \left(37 + 6\,x - 9\,x^{2} + x^{3} - 12\,n - 12\,n x + 3\,n x^{2} - 3\,n^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n-3}(x) + \left(-10 + 15\,x - 3\,x^{2} + 12\,n - 6\,n x - 3\,n^{2}\right)\,P_{n-2}(x) + \left(-6 + 3\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 3 + 4\,x\)
\(P_{4}(x) = 8 + 24\,x + 9\,x^{2}\)
\(P_{5}(x) = 24 + 116\,x + 90\,x^{2} + 16\,x^{3}\)
\(P_{6}(x) = 89 + 580\,x + 675\,x^{2} + 240\,x^{3} + 25\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3347
4824941
5241169016246
689580675240251609
74153256490525605253611697
82372209723761125200752510084994737

All 15 rows

Technical details
Node id
n-82918093e705d397 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-35acbe4538fbbff2

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      0;
   2:      1;
   3:      3,       4;
   4:      8,      24,       9;
   5:     24,     116,      90,      16;
   6:     89,     580,     675,     240,      25;
   7:    415,    3256,    4905,    2560,     525,     36;
   8:   2372,   20972,   37611,   25200,    7525,   1008,    49;
   9:  16072,  154472,  313740,  250544,   95900,  18648,  1764,   64;
  10: 125673, 1286760, 2870334, 2611056, 1194375, 297864, 40866, 2880, 81;
%F Formula
The row polynomials satisfy P(n,x) = -(n^3-15*n^2+74*n-120)*P(n-6,x) + (3*n^3-39*n^2+168*n-240+(3*n^2-27*n+60)*x)*P(n-5,x) - (3*n^3-30*n^2+97*n-100+(6*n^2-45*n+84)*x+(3*n-12)*x^2)*P(n-4,x) + (n^3-3*n^2-12*n+37+(3*n^2-12*n+6)*x+(3*n-9)*x^2+x^3)*P(n-3,x) - (3*n^2-12*n+10+(6*n-15)*x+3*x^2)*P(n-2,x) + (3*n-6+3*x)*P(n-1,x) for n >= 7, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 3 + 4*x, P(4,x) = 8 + 24*x + 9*x^2, P(5,x) = 24 + 116*x + 90*x^2 + 16*x^3, P(6,x) = 89 + 580*x + 675*x^2 + 240*x^3 + 25*x^4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 3 + 4*x;
P[4, x_] := 8 + 24*x + 9*x^2;
P[5, x_] := 24 + 116*x + 90*x^2 + 16*x^3;
P[6, x_] := 89 + 580*x + 675*x^2 + 240*x^3 + 25*x^4;
P[n_, x_] := P[n, x] = Expand[-(n^3 - 15*n^2 + 74*n - 120)*P[n - 6, x] + (3*n^3 - 39*n^2 + 168*n - 240 + (3*n^2 - 27*n + 60)*x)*P[n - 5, x] - (3*n^3 - 30*n^2 + 97*n - 100 + (6*n^2 - 45*n + 84)*x + (3*n - 12)*x^2)*P[n - 4, x] + (n^3 - 3*n^2 - 12*n + 37 + (3*n^2 - 12*n + 6)*x + (3*n - 9)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 12*n + 10 + (6*n - 15)*x + 3*x^2)*P[n - 2, x] + (3*n - 6 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 3 + 4*x; v[4] = 8 + 24*x + 9*x^2; v[5] = 24 + 116*x + 90*x^2 + 16*x^3; v[6] = 89 + 580*x + 675*x^2 + 240*x^3 + 25*x^4; for(n = 7, N, my(i = n); v[i] = -(n^3 - 15*n^2 + 74*n - 120)*v[i - 6] + (3*n^3 - 39*n^2 + 168*n - 240 + (3*n^2 - 27*n + 60)*x)*v[i - 5] - (3*n^3 - 30*n^2 + 97*n - 100 + (6*n^2 - 45*n + 84)*x + (3*n - 12)*x^2)*v[i - 4] + (n^3 - 3*n^2 - 12*n + 37 + (3*n^2 - 12*n + 6)*x + (3*n - 9)*x^2 + x^3)*v[i - 3] - (3*n^2 - 12*n + 10 + (6*n - 15)*x + 3*x^2)*v[i - 2] + (3*n - 6 + 3*x)*v[i - 1]); 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

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