Real-rooted atlas

A207607 → basis A007318 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−6, −2)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(8 - 16\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(22 + 4\,x - 35\,n - 8\,n x + 13\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-30 - 26\,x - 4\,x^{2} + 48\,n + 32\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(36 + 24\,x + 3\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
21913234
37169212163
4265334151292781
598915329172653723742
6369167835049194841145217929
713775292912606212576355558953285903

All 15 rows

Technical details
Node id
n-28552443aa1be378 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-57c034ccac65b14f

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:     19,     13,      2;
  3:     71,     69,     21,      2;
  4:    265,    334,    151,     29,      2;
  5:    989,   1532,    917,    265,     37,     2;
  6:   3691,   6783,   5049,   1948,    411,    45,    2;
  7:  13775,  29291,  26062,  12576,   3555,   589,   53,    2;
  8:  51409, 124156, 128490,  74418,  26385,  5866,  799,   61,  2;
  9: 191861, 518742, 612054, 413586, 176403, 49260, 9009, 1041, 69, 2;
%F Formula
The row polynomials satisfy (8*n^2-16*n+8)*P(n,x) = (13*n^2-35*n+22+(4*n^2-8*n+4)*x)*P(n-1,x) + (48*n-30+(32*n-26)*x+(4*n-4)*x^2)*d/dx P(n-1,x) + (36+24*x+3*x^2)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((13*n^2 - 35*n + 22 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] + (48*n - 30 + (32*n - 26)*x + (4*n - 4)*x^2)*D[P[n - 1, x], x] + (36 + 24*x + 3*x^2)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 16*n + 8)]];
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] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((13*n^2 - 35*n + 22 + (4*n^2 - 8*n + 4)*x)*v[i - 1] + (48*n - 30 + (32*n - 26)*x + (4*n - 4)*x^2)*deriv(v[i - 1], x) + (36 + 24*x + 3*x^2)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 16*n + 8)); 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

Candidate equalities ?

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023