Real-rooted atlas

A116412 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(20 + 4\,x - 13\,n - 2\,n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(84 + 48\,x + 5\,x^{2} - 24\,n - 12\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(48 + 48\,x + 14\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1516
22210133
39869151183
44364161412011014
51940233610792382515619
68632125527350221236030131137
7384086545246446177533940507351172542

All 15 rows

Technical details
Node id
n-29c8bdd35d90da39 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8093bc57471dc962

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:      5,       1;
  2:     22,      10,       1;
  3:     98,      69,      15,      1;
  4:    436,     416,     141,     20,      1;
  5:   1940,    2336,    1079,    238,     25,     1;
  6:   8632,   12552,    7350,   2212,    360,    30,    1;
  7:  38408,   65452,   46446,  17753,   3940,   507,   35,   1;
  8: 170896,  333952,  278488, 129232,  36445,  6388,  679,  40,  1;
  9: 760400, 1676016, 1606248, 877368, 300645, 66951, 9681, 876, 45, 1;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (3*n^2-13*n+20-(2*n-4)*x)*P(n-1,x) - (24*n-84+(12*n-48)*x+(n-5)*x^2)*d/dx P(n-1,x) + (48+48*x+14*x^2+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[((3*n^2 - 13*n + 20 - (2*n - 4)*x)*P[n - 1, x] - (24*n - 84 + (12*n - 48)*x + (n - 5)*x^2)*D[P[n - 1, x], x] + (48 + 48*x + 14*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n + 1)]];
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^2 - 13*n + 20 - (2*n - 4)*x)*v[i - 1] - (24*n - 84 + (12*n - 48)*x + (n - 5)*x^2)*deriv(v[i - 1], x) + (48 + 48*x + 14*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 - 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