Real-rooted atlas

A094816 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
31918441
4106142608316
56851152692176162721
6505610018760027604803225946
7420799451484976391609760124864271801
83904549691909931885449921730403180831361283105936

All 15 rows

Technical details
Node id
n-37d14fc6e26a45d7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f9ffb7958d11888e

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:       19,        18,         4;
   4:      106,       142,        60,         8;
   5:      685,      1152,       692,       176,       16;
   6:     5056,     10018,      7600,      2760,      480,       32;
   7:    42079,     94514,     84976,     39160,     9760,     1248,      64;
   8:   390454,    969190,    993188,    544992,   173040,    31808,    3136,    128;
   9:  4000441,  10780400,  12268616,   7707168,  2925104,   687232,   97664,   7680,   256;
  10: 44881660, 129612162, 160838688, 112663312, 49131264, 13842528, 2521344, 286464, 18432, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2)*P(n-2,x) + (n+2+2*x)*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[-(n - 2)*P[n - 2, x] + (n + 2 + 2*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 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*v[i - 2] + (n + 2 + 2*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

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