Real-rooted atlas

A033282 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−3, −2)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(10 + 4\,x + 5\,n + 2\,n x\right)\,P_{n-1}(x) + \left(12 + 10\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
23125561
321526110514595
4159725901568420426217
51242525220204008220165013268047
699955243675246735132825400956435429770149
78246752346995285477519238457757751871872502514308939707

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(2 + n\right)\,P_{n}(x) = \left(1 - n\right)\,P_{n-2}(x) + \left(5 + 2\,x + 10\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)
Technical details
Node id
n-fb13dff3dfb4d666 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-fdd65bc6f0fa1af3)
Fingerprint
f6-78576ef9a6880520

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:        5,         2;
  2:       31,        25,         5;
  3:      215,       261,       105,        14;
  4:     1597,      2590,      1568,       420,        42;
  5:    12425,     25220,     20400,      8220,      1650,      132;
  6:    99955,    243675,    246735,    132825,     40095,     6435,      429;
  7:   824675,   2346995,   2854775,   1923845,    775775,   187187,    25025,    1430;
  8:  6939769,  22582780,  32072656,  25965940,  13107094,  4224220,   848848,   97240,   4862;
  9: 59334605, 217299942, 352928160, 333648588, 202333950, 81624816, 21905520, 3771144, 377910, 16796;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (5*n+10+(2*n+4)*x)*P(n-1,x) + (12+10*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 10 + (2*n + 4)*x)*P[n - 1, x] + (12 + 10*x + 2*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n + 10 + (2*n + 4)*x)*v[i - 1] + (12 + 10*x + 2*x^2)*deriv(v[i - 1], x))/(n + 2)); 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