Real-rooted atlas

A166341 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11212125
24872261147
32404802945411069
4144036003096104411019291
5100803024033480165603462222194045
680640282240382320250200800881105244611086987
7725760290304046519203795120165144036456034494894114127229
872576003265920060439680591292803267936010080000159621610616417901203949291

All 15 rows

Technical details
Node id
n-4654cca59858ef1f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b6466966904f5921

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:        12,         12,           1;
   2:        48,         72,          26,           1;
   3:       240,        480,         294,          54,           1;
   4:      1440,       3600,        3096,        1044,         110,          1;
   5:     10080,      30240,       33480,       16560,        3462,        222,          1;
   6:     80640,     282240,      382320,      250200,       80088,      11052,        446,         1;
   7:    725760,    2903040,     4651920,     3795120,     1651440,     364560,      34494,       894,        1;
   8:   7257600,   32659200,    60439680,    59129280,    32679360,   10080000,    1596216,    106164,     1790,      1;
   9:  79833600,  399168000,   837889920,   956551680,   642660480,  256556160,   58381080,   6809520,   323862,   3582,    1;
  10: 958003200, 5269017600, 12370579200, 16149974400, 12793697280, 6294516480, 1889623440, 325953000, 28533528, 982332, 7166, 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) = 12 + 12*x + x^2.
%t Mathematica
P[1, x_] := 12 + 12*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] = 12 + 12*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