Real-rooted atlas

A008288 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(15 + 3\,x - 10\,n - n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(64 + 41\,x + 5\,x^{2} - 17\,n - 10\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(34 + 37\,x + 12\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
3117119
4393711188
513917579151409
64957774711371911900
71763331525239912112318827
8627913757126036241179930127141008

All 15 rows

Technical details
Node id
n-77c49e527abe45d3 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-72610dbf058ffa6b

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     3,      1;
   3:    11,      7,      1;
   4:    39,     37,     11,      1;
   5:   139,    175,     79,     15,     1;
   6:   495,    777,    471,    137,    19,     1;
   7:  1763,   3315,   2523,    991,   211,    23,    1;
   8:  6279,  13757,  12603,   6241,  1799,   301,   27,   1;
   9: 22363,  55943,  59927,  35831, 13051,  2959,  407,  31,  1;
  10: 79647, 223985, 274687, 192505, 84823, 24329, 4535, 529, 35, 1;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n^2-10*n+15-(n-3)*x)*P(n-1,x) - (17*n-64+(10*n-41)*x+(n-5)*x^2)*d/dx P(n-1,x) + (34+37*x+12*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 10*n + 15 - (n - 3)*x)*P[n - 1, x] - (17*n - 64 + (10*n - 41)*x + (n - 5)*x^2)*D[P[n - 1, x], x] + (34 + 37*x + 12*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 10*n + 15 - (n - 3)*x)*v[i - 1] - (17*n - 64 + (10*n - 41)*x + (n - 5)*x^2)*deriv(v[i - 1], x) + (34 + 37*x + 12*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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