Real-rooted atlas

A207606 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(18 + 24\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(81 + 81\,x + 18\,x^{2} + 120\,n + 126\,n x + 30\,n x^{2} + 44\,n^{2} + 48\,n^{2} x + 12\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-57 - 126\,x - 87\,x^{2} - 18\,x^{3} - 50\,n - 120\,n x - 94\,n x^{2} - 24\,n x^{3}\right)\,P_{n-1}'(x) + \left(15 + 57\,x + 81\,x^{2} + 51\,x^{3} + 12\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 13 + 15\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01315432
13458315128
289201162546512
32336557073618572048
461020522778191870112588192
51597625510188888044251239175932768
64181186873558237490237209122204423610131072
710946549741198251481231137675583717298319830911524288

All 15 rows

Technical details
Node id
n-45fb91dbddcb6bad · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d049d83a04ac9840

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:    13,     15,       4;
  1:    34,     58,      31,       5;
  2:    89,    201,     162,      54,       6;
  3:   233,    655,     707,     361,      85,       7;
  4:   610,   2052,    2778,    1918,     701,     125,      8;
  5:  1597,   6255,   10188,    8880,    4425,    1239,    175,      9;
  6:  4181,  18687,   35582,   37490,   23720,    9122,   2044,    236,    10;
  7: 10946,  54974,  119825,  148123,  113767,   55837,  17298,   3198,   309,   11;
  8: 28657, 159765,  392286,  556678,  503265,  300993, 119560,  30744,  4797,  395,  12;
  9: 75025, 459743, 1255669, 2011859, 2093313, 1478015, 717927, 237721, 51876, 6952, 495, 13;
%F Formula
The row polynomials satisfy (8*n^2+24*n+18)*P(n,x) = (44*n^2+120*n+81+(48*n^2+126*n+81)*x+(12*n^2+30*n+18)*x^2)*P(n-1,x) - (50*n+57+(120*n+126)*x+(94*n+87)*x^2+(24*n+18)*x^3)*d/dx P(n-1,x) + (15+57*x+81*x^2+51*x^3+12*x^4)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 13 + 15*x + 4*x^2.
%t Mathematica
P[0, x_] := 13 + 15*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((44*n^2 + 120*n + 81 + (48*n^2 + 126*n + 81)*x + (12*n^2 + 30*n + 18)*x^2)*P[n - 1, x] - (50*n + 57 + (120*n + 126)*x + (94*n + 87)*x^2 + (24*n + 18)*x^3)*D[P[n - 1, x], x] + (15 + 57*x + 81*x^2 + 51*x^3 + 12*x^4)*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] = 13 + 15*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((44*n^2 + 120*n + 81 + (48*n^2 + 126*n + 81)*x + (12*n^2 + 30*n + 18)*x^2)*v[i - 1] - (50*n + 57 + (120*n + 126)*x + (94*n + 87)*x^2 + (24*n + 18)*x^3)*deriv(v[i - 1], x) + (15 + 57*x + 81*x^2 + 51*x^3 + 12*x^4)*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

Same fingerprint ?

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