Real-rooted atlas

A078812 → x ↦ x+1 → basis A013611

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2448
315321663
45618419264496
5209928148810242563905
67804364947299845120102430744
7291119648538567680060160245764096242047
81086485872284544512640542720337920114688163841905632

All 15 rows

Technical details
Node id
n-41513f0d6e6cd398 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-cf9b351a67dba233

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,       4;
   3:     15,      32,      16;
   4:     56,     184,     192,       64;
   5:    209,     928,    1488,     1024,      256;
   6:    780,    4364,    9472,     9984,     5120,     1024;
   7:   2911,   19648,   53856,    76800,    60160,    24576,     4096;
   8:  10864,   85872,  284544,   512640,   542720,   337920,   114688,   16384;
   9:  40545,  367296, 1427808,  3111936,  4161280,  3497984,  1806336,  524288,   65536;
  10: 151316, 1545492, 6895872, 17646336, 28550144, 30299136, 21102592, 9306112, 2359296, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 4*x)*P[n - 1, x]];
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 + 4*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (4 + 4*x)*v[i - 1]); 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