Real-rooted atlas

A166344 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 8 + 8\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
188117
2324818199
3160320198381717
4960240020727087816219
567202016022360111202358158162877
6537601881602551201674005391275483181726219
748384019353603102960253512011065602458402359863819433917
8483840021772800403065603946656021850080676200010777527270812781136148139

All 15 rows

Technical details
Node id
n-25a1bf976309f071 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ea33b747599ac881

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 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:         8,          8,          1;
   2:        32,         48,         18,           1;
   3:       160,        320,        198,          38,          1;
   4:       960,       2400,       2072,         708,         78,          1;
   5:      6720,      20160,      22360,       11120,       2358,        158,          1;
   6:     53760,     188160,     255120,      167400,      53912,       7548,        318,         1;
   7:    483840,    1935360,    3102960,     2535120,    1106560,     245840,      23598,       638,        1;
   8:   4838400,   21772800,   40306560,    39466560,   21850080,    6762000,    1077752,     72708,     1278,      1;
   9:  53222400,  266112000,  558714240,   638184960,  429216480,  171672480,   39198760,   4601840,   221958,   2558,    1;
  10: 638668800, 3512678400, 8248262400, 10772092800, 8539211520, 4206222720, 1265227440, 219003000, 19295192, 673548, 5118, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+2+(n+1)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 8 + 8*x + x^2.
%t Mathematica
P[1, x_] := 8 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + 2 + (n + 1)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 8 + 8*x + x^2; for(n = 2, N, my(i = n); v[i] = (n + 2 + (n + 1)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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. [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