Real-rooted atlas

A035607 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 - 5\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(47 + 31\,x + 4\,x^{2} - 17\,n - 10\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(34 + 37\,x + 12\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2527
31712231
46162202145
5217288138282673
677312668042463623127
7275353644242173638644214527

All 15 rows

Technical details
Node id
n-d87adca1089fcb13 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-48a53241c969ef5c

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:     1;
  2:     5,     2;
  3:    17,    12,     2;
  4:    61,    62,    20,     2;
  5:   217,   288,   138,    28,     2;
  6:   773,  1266,   804,   246,    36,    2;
  7:  2753,  5364,  4242,  1736,   386,   44,   2;
  8:  9805, 22150, 20964, 10746,  3212,  558,  52,  2;
  9: 34921, 89736, 98890, 60916, 22890, 5360, 762, 60, 2;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (2*n^2-5*n+2)*P(n-1,x) - (17*n-47+(10*n-31)*x+(n-4)*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 >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 2)*P[n - 1, x] - (17*n - 47 + (10*n - 31)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (34 + 37*x + 12*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 2)]];
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] = 1; v[3] = 5 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 2)*v[i - 1] - (17*n - 47 + (10*n - 31)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (34 + 37*x + 12*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 2)); 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