Real-rooted atlas

A257623 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 2\,x + 5\,n + 5\,n x\right)\,P_{n-1}(x) + \left(-5\,x - 5\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
266669141
310561584582273249
422176443522699448188198421
557657614414401230240403920390302433691449
6178738565362156858988160286070405680290313698729165085341
764345881622521058563035096064195747552060581136077294448251395821878573758209

All 15 rows

Technical details
Node id
n-f36340a6b18e36e9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-04577fbe25f76a46

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:            66,            66,              9;
  3:          1056,          1584,            582,            27;
  4:         22176,         44352,          26994,          4818,            81;
  5:        576576,       1441440,        1230240,        403920,         39030,           243;
  6:      17873856,      53621568,       58988160,      28607040,       5680290,        313698,          729;
  7:     643458816,    2252105856,     3035096064,    1957475520,     605811360,      77294448,      2513958,        2187;
  8:   26381811456,  105527245824,   168407471232,  135877053312,   57743975520,   12141315648,   1032481362,    20124786,      6561;
  9: 1213563326976, 5461034971392, 10072704305664, 9769635203328, 5305900856256, 1583079065568, 235063383456, 13643630352, 161037654, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+1+(5*n-2)*x)*P(n-1,x) - (5*x+5*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[(5*n + 1 + (5*n - 2)*x)*P[n - 1, x] - (5*x + 5*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] = (5*n + 1 + (5*n - 2)*x)*v[i - 1] - (5*x + 5*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