Real-rooted atlas

A207824 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−9, −5)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(7 + x + 7\,n + n x\right)\,P_{n-1}(x) + \left(45 + 14\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1718
24814163
3329145211496
4225513302912813905
51545611420334648635130744
610593794066345516720730421242047
7726103752979332577811051179510234911905632
849767845902890304646789359216294018914136556115003009
934111385455440352689558292205062022377294315284201756631118118440
1023380291134701674023076664290545532232142054063668491890406562196701929944511
111602508992261737605619354876528553648602510225905136556674784787747265596526857717321437648
121098376003319577124644159350229787832496140258930878560651788410322302212860904116428574690322384157641556673
1375283811239145406256485129186797838699071310982570663504567835684071321600560192474624209549341684430971743810911453811015736
14516002918640107355048199010337788183736107042193842472642676287258509601015931549305265606202433799487732671254236142512376044469813572846569215

Other recurrences

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

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(7 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + x\)
Technical details
Node id
n-3c7673a6eb830856 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-08d21cc057ab3854)
Fingerprint
f6-1298dcd5b9dfe51d

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:        7,        1;
  2:       48,       14,        1;
  3:      329,      145,       21,       1;
  4:     2255,     1330,      291,      28,       1;
  5:    15456,    11420,     3346,     486,      35,      1;
  6:   105937,    94066,    34551,    6720,     730,     42,     1;
  7:   726103,   752979,   332577,   81105,   11795,   1023,    49,    1;
  8:  4976784,  5902890,  3046467,  893592,  162940,  18914,  1365,   56,  1;
  9: 34111385, 45544035, 26895582, 9220506, 2022377, 294315, 28420, 1756, 63, 1;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (7*n+7+(n+1)*x)*P(n-1,x) + (45+14*x+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[((7*n + 7 + (n + 1)*x)*P[n - 1, x] + (45 + 14*x + x^2)*D[P[n - 1, x], x])/(2*n)]];
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] = ((7*n + 7 + (n + 1)*x)*v[i - 1] + (45 + 14*x + x^2)*deriv(v[i - 1], x))/(2*n)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023