Real-rooted atlas

A257606 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + 3\,x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
2727216160
37201080488642352
479201584010872295225637840
59504023760021960091800168081024661872
6123552037065604316400245520070207292232409612512080
71729728060540480851558406153840024207120504252049392816384254291952

All 15 rows

Technical details
Node id
n-8e2b4e2d6609e6fb · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-79e4efb037c60552

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 0)
  • 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:          1;
  1:          8,           4;
  2:         72,          72,          16;
  3:        720,        1080,         488,          64;
  4:       7920,       15840,       10872,        2952,         256;
  5:      95040,      237600,      219600,       91800,       16808,       1024;
  6:    1235520,     3706560,     4316400,     2455200,      702072,      92232,       4096;
  7:   17297280,    60540480,    85155840,    61538400,    24207120,    5042520,     493928,     16384;
  8:  259459200,  1037836800,  1712430720,  1504863360,   758585520,  219875040,   34700472,   2600712,    65536;
  9: 4151347200, 18681062400, 35390234880, 36687530880, 22646796480, 8487309600, 1886130000, 231609240, 13527848, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (n+7+(n+3)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + 7 + (n + 3)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + 7 + (n + 3)*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 - 1, v[i])))};
T(9)

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