Real-rooted atlas

A327997 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-54 + 45\,n - 9\,n^{2}\right)\,P_{n-4}(x) + \left(-18 - 12\,x - 3\,n + 6\,n x + 6\,n^{2}\right)\,P_{n-3}(x) + \left(5 - 4\,x - x^{2} - 10\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(5 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
\(P_{2}(x) = 17 + 18\,x + 3\,x^{2}\)
\(P_{3}(x) = 78 + 130\,x + 45\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21718338
378130454257
43939044838851873
52208645846801300150614802
61397748978445621662029002347127278
710002640149043484720221647075571234381191717

All 15 rows

Technical details
Node id
n-5593784c7b4efd41 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-bd75928729d07b6b

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:       4,        2;
  2:      17,       18,        3;
  3:      78,      130,       45,        4;
  4:     393,      904,      483,       88,        5;
  5:    2208,     6458,     4680,     1300,      150,       6;
  6:   13977,    48978,    44562,    16620,     2900,     234,      7;
  7:  100026,   401490,   434847,   202216,    47075,    5712,    343,     8;
  8:  806769,  3587904,  4449750,  2455152,   709695,  114408,  10290,   480,   9;
  9: 7280604, 35032626, 48342528, 30541640, 10455480, 2087442, 248724, 17328, 648, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-3*n-18+(6*n-12)*x)*P(n-3,x) - (n^2+10*n-5+(2*n+4)*x+x^2)*P(n-2,x) + (2*n+5+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 17 + 18*x + 3*x^2, P(3,x) = 78 + 130*x + 45*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[2, x_] := 17 + 18*x + 3*x^2;
P[3, x_] := 78 + 130*x + 45*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 3*n - 18 + (6*n - 12)*x)*P[n - 3, x] - (n^2 + 10*n - 5 + (2*n + 4)*x + x^2)*P[n - 2, x] + (2*n + 5 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; v[3] = 17 + 18*x + 3*x^2; v[4] = 78 + 130*x + 45*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 3*n - 18 + (6*n - 12)*x)*v[i - 3] - (n^2 + 10*n - 5 + (2*n + 4)*x + x^2)*v[i - 2] + (2*n + 5 + 2*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 - 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