Real-rooted atlas

A257625 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3\,x + 6\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-6\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
272729153
312961944702273969
4311046220837584648081137457
593312023328001982880641520588062435949369
6335923201007769601105747205318784010328472530712729308991753
7141087744049380710406642881280426202560013028571361612956244780782218718722791089

All 15 rows

Technical details
Node id
n-7bc03246f43906fd · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2a2e8db5a042c827

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:             6,              3;
  2:            72,             72,              9;
  3:          1296,           1944,            702,             27;
  4:         31104,          62208,          37584,           6480,             81;
  5:        933120,        2332800,        1982880,         641520,          58806,           243;
  6:      33592320,      100776960,      110574720,       53187840,       10328472,        530712,          729;
  7:    1410877440,     4938071040,     6642881280,     4262025600,     1302857136,     161295624,      4780782,        2187;
  8:   67722117120,   270888468480,   431728496640,   347075850240,   146343262464,   30263321088,   2476803744,    43040160,      6561;
  9: 3656994324480, 16456474460160, 30322577940480, 29332141977600, 15843800931840, 4677587792640, 680112210240, 37668538080, 387400806, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (6*n+(6*n-3)*x)*P(n-1,x) - (6*x+6*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[(6*n + (6*n - 3)*x)*P[n - 1, x] - (6*x + 6*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] = (6*n + (6*n - 3)*x)*v[i - 1] - (6*x + 6*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