Real-rooted atlas

A144439 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21010121
36496341195
4532106463810612341
5544813620115363684322134611
666440199320218272104344199229701609269
79410563293696442555228296408579521041362914112454947

All 15 rows

Technical details
Node id
n-16dc446fc8a01758 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-42d134b85075e1e7

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:        64,         96,         34,          1;
  4:       532,       1064,        638,        106,          1;
  5:      5448,      13620,      11536,       3684,        322,         1;
  6:     66440,     199320,     218272,     104344,      19922,       970,        1;
  7:    941056,    3293696,    4425552,    2829640,     857952,    104136,     2914,       1;
  8:  15189776,   60759104,   96759856,   77622704,   32578908,   6672264,   534278,    8746,     1;
  9: 275298080, 1238841360, 2281463040, 2203860960, 1187014008, 348512460, 50125616, 2712204, 26242, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + (2*n+(2*n-1)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + (2*n + (2*n - 1)*x)*P[n - 1, x] - (2*x + 2*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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + (2*n + (2*n - 1)*x)*v[i - 1] - (2*x + 2*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