Real-rooted atlas

A094816 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 18\,x - 9\,n + 6\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(4 + 6\,x - 9\,x^{2} - 6\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-1 + 6\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 6\,x\)
\(P_{3}(x) = 5 + 30\,x + 27\,x^{2}\)
\(P_{4}(x) = 16 + 138\,x + 243\,x^{2} + 108\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2268
35302762
416138243108505
565696175515124054433
6326401412690156608100145842248
719572657498118152820113400393665103437338
81370020025082838715210721431675714420178605174964905605

All 15 rows

Technical details
Node id
n-ec382b0d6dcc0eac · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ed4c9b6d0be1e5df

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 -1)
  • 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:      1;
   2:      2,        6;
   3:      5,       30,       27;
   4:     16,      138,      243,       108;
   5:     65,      696,     1755,      1512,       405;
   6:    326,     4014,    12690,     15660,      8100,      1458;
   7:   1957,    26574,    98118,    152820,    113400,     39366,     5103;
   8:  13700,   200250,   828387,   1521072,   1431675,    714420,   178605,    17496;
   9: 109601,  1698432,  7669782,  15933456,  17795295,  11308248,  4072194,   769824,   59049;
  10: 986410, 16039926, 77713668, 177845112, 226250010, 171430182, 78871968, 21520080, 3188646, 196830;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(6*n-18)*x)*P(n-3,x) - (n^2-4+(6*n-6)*x+9*x^2)*P(n-2,x) + (2*n-1+6*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 6*x, P(3,x) = 5 + 30*x + 27*x^2, P(4,x) = 16 + 138*x + 243*x^2 + 108*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 6*x;
P[3, x_] := 5 + 30*x + 27*x^2;
P[4, x_] := 16 + 138*x + 243*x^2 + 108*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (6*n - 18)*x)*P[n - 3, x] - (n^2 - 4 + (6*n - 6)*x + 9*x^2)*P[n - 2, x] + (2*n - 1 + 6*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] = 1; v[2] = 2 + 6*x; v[3] = 5 + 30*x + 27*x^2; v[4] = 16 + 138*x + 243*x^2 + 108*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (6*n - 18)*x)*v[i - 3] - (n^2 - 4 + (6*n - 6)*x + 9*x^2)*v[i - 2] + (2*n - 1 + 6*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