Real-rooted atlas

A207606 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(18 + 24\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(36 + 12\,x + 42\,n + 14\,n x + 13\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(108 + 57\,x + 6\,x^{2} + 48\,n + 32\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(36 + 24\,x + 3\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 26 + 15\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02615243
19784232206
2362418174312987
31351195010912963924729
4504287336140224445047222658
5188173802432202148204005636552108561
67022616218016069289238303906502854632520147
726208768092277274650282420679355762986311047122492174

All 15 rows

Technical details
Node id
n-6872f64788e7602d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-34678cae11e60551

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:      26,       15,        2;
  1:      97,       84,       23,        2;
  2:     362,      418,      174,       31,       2;
  3:    1351,     1950,     1091,      296,      39,       2;
  4:    5042,     8733,     6140,     2244,     450,      47,      2;
  5:   18817,    38024,    32202,    14820,    4005,     636,     55,      2;
  6:   70226,   162180,   160692,    89238,   30390,    6502,    854,     63,     2;
  7:  262087,   680922,   772746,   502824,  206793,   55762,   9863,   1104,    71,    2;
  8:  978122,  2823595,  3611214,  2694804, 1299606,  423339,  94360,  14216,  1386,   79,  2;
  9: 3650401, 11591580, 16495705, 13887606, 7686435, 2937200, 790916, 150120, 19689, 1700, 87, 2;
%F Formula
The row polynomials satisfy (8*n^2+24*n+18)*P(n,x) = (13*n^2+42*n+36+(4*n^2+14*n+12)*x)*P(n-1,x) + (48*n+108+(32*n+57)*x+(4*n+6)*x^2)*d/dx P(n-1,x) + (36+24*x+3*x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 26 + 15*x + 2*x^2.
%t Mathematica
P[0, x_] := 26 + 15*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((13*n^2 + 42*n + 36 + (4*n^2 + 14*n + 12)*x)*P[n - 1, x] + (48*n + 108 + (32*n + 57)*x + (4*n + 6)*x^2)*D[P[n - 1, x], x] + (36 + 24*x + 3*x^2)*D[P[n - 1, x], {x, 2}])/(8*n^2 + 24*n + 18)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 26 + 15*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((13*n^2 + 42*n + 36 + (4*n^2 + 14*n + 12)*x)*v[i - 1] + (48*n + 108 + (32*n + 57)*x + (4*n + 6)*x^2)*deriv(v[i - 1], x) + (36 + 24*x + 3*x^2)*deriv(deriv(v[i - 1], x), x))/(8*n^2 + 24*n + 18)); 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