Real-rooted atlas

A142458 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21010121
380120421243
48801760105017013861
51232030800260008200682178003
620944062832068640032560060810273011913301
74188800146608001966272012504800374248043932010922155209843

All 15 rows

Technical details
Node id
n-0b7c6f7d1239dc9e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-72ea6b42576fa2d7

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:          2,           1;
  2:         10,          10,           1;
  3:         80,         120,          42,           1;
  4:        880,        1760,        1050,         170,           1;
  5:      12320,       30800,       26000,        8200,         682,          1;
  6:     209440,      628320,      686400,      325600,       60810,       2730,         1;
  7:    4188800,    14660800,    19662720,    12504800,     3742480,     439320,     10922,        1;
  8:   96342400,   385369600,   612821440,   489670720,   203729680,   40939360,   3129850,    43690,      1;
  9: 2504902400, 11272060800, 20734560000, 19968009600, 10686947040, 3098818800, 434432400, 22127400, 174762, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-1+(3*n-2)*x)*P(n-1,x) - (3*x+3*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[(3*n - 1 + (3*n - 2)*x)*P[n - 1, x] - (3*x + 3*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] = (3*n - 1 + (3*n - 2)*x)*v[i - 1] - (3*x + 3*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