Real-rooted atlas

A086645 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-20 - 12\,x + 10\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-6 - 10\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
32824456
4208264968576
5155226241504320166016
611584244801920067209603262976
78646421926421830410675225920268864659456
86453761909376230451214488324999689049671681286905856

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-4 - 8\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)
Technical details
Node id
n-6a5cb0b957c123da · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-2729d81eb6e0aef8)
Fingerprint
f6-73a02a3902b59fb9

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        4,         2;
   3:       28,        24,         4;
   4:      208,       264,        96,         8;
   5:     1552,      2624,      1504,       320,        16;
   6:    11584,     24480,     19200,      6720,       960,       32;
   7:    86464,    219264,    218304,    106752,     25920,     2688,      64;
   8:   645376,   1909376,   2304512,   1448832,    499968,    90496,    7168,    128;
   9:  4817152,  16287744,  23100416,  17758208,   7964160,  2078720,  293888,  18432,   256;
  10: 35955712, 136770048, 222879744, 202598400, 111937536, 38329344, 7913472, 903168, 46080, 512;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (10*n-20+(6*n-12)*x)*P(n-1,x) - (6+10*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((10*n - 20 + (6*n - 12)*x)*P[n - 1, x] - (6 + 10*x + 4*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = ((10*n - 20 + (6*n - 12)*x)*v[i - 1] - (6 + 10*x + 4*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, v[i])))};
T(10)

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