Real-rooted atlas

A114656 → θ+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(2 + 2\,x - 7\,n - 5\,n x + 5\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-1 + x^{2} + 2\,n - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
31918340
496132484280
550190051010052016
62668591045601460180614784
71440737926370651666034652947109824
8785922396242835841649204928072244488823680

All 15 rows

Technical details
Node id
n-3314ca1e7e4eb797 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bf35de5d7524cff0

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:       4,       2;
   3:      19,      18,        3;
   4:      96,     132,       48,        4;
   5:     501,     900,      510,      100,       5;
   6:    2668,    5910,     4560,     1460,     180,       6;
   7:   14407,   37926,    37065,    16660,    3465,     294,      7;
   8:   78592,  239624,   283584,   164920,   49280,    7224,    448,     8;
   9:  432073, 1497096,  2080764,  1488312,  585270,  125496,  13692,   648,   9;
  10: 2390004, 9274410, 14806080, 12585720, 6166440, 1764252, 285600, 24120, 900, 10;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (5*n^2-7*n+2+(3*n^2-5*n+2)*x)*P(n-1,x) + (2*n-1-(2*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 7*n + 2 + (3*n^2 - 5*n + 2)*x)*P[n - 1, x] + (2*n - 1 - (2*n - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((5*n^2 - 7*n + 2 + (3*n^2 - 5*n + 2)*x)*v[i - 1] + (2*n - 1 - (2*n - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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

Same fingerprint ?

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