Real-rooted atlas

A081580 → basis A013611 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(42 + 16\,x - 42\,n - 8\,n x + 12\,n^{2}\right)\,P_{n-1}(x) + \left(188 + 190\,x + 32\,x^{2} - 68\,n - 64\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(85 + 148\,x + 74\,x^{2} + 8\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2616816145
3516872448641900
44621102167936249625625525
54062611190011712057280128001024340750
63592811181724155304010332803622406246440964556125
73171336121374881921113616186240777216021053442949121638460895000

All 15 rows

Technical details
Node id
n-2bd60c45ab7c2c34 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-31b02e78b9a6fd3c

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,          4;
  2:        61,         68,         16;
  3:       516,        872,        448,         64;
  4:      4621,      10216,       7936,       2496,        256;
  5:     40626,     111900,     117120,      57280,      12800,       1024;
  6:    359281,    1181724,    1553040,    1033280,     362240,      62464,      4096;
  7:   3171336,   12137488,   19211136,   16186240,    7772160,    2105344,    294912,    16384;
  8:  28010041,  122238992,  226277248,  230854784,  141099520,   52527104,  11542528,  1359872,   65536;
  9: 247343646, 1212338036, 2569647616, 3079116416, 2288045056, 1087637504, 328843264, 60637184, 6160384, 262144;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (12*n^2-42*n+42-(8*n-16)*x)*P(n-1,x) - (68*n-188+(64*n-190)*x+(8*n-32)*x^2)*d/dx P(n-1,x) + (85+148*x+74*x^2+8*x^3)*d^2/dx^2 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[Together[((12*n^2 - 42*n + 42 - (8*n - 16)*x)*P[n - 1, x] - (68*n - 188 + (64*n - 190)*x + (8*n - 32)*x^2)*D[P[n - 1, x], x] + (85 + 148*x + 74*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(2*n)]];
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] = ((12*n^2 - 42*n + 42 - (8*n - 16)*x)*v[i - 1] - (68*n - 188 + (64*n - 190)*x + (8*n - 32)*x^2)*deriv(v[i - 1], x) + (85 + 148*x + 74*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n)); 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

Candidate equalities ?

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