Real-rooted atlas

A035607 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(10 + 16\,x + 4\,x^{2} - 9\,n - 12\,n x - 2\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(28 + 59\,x + 36\,x^{2} + 5\,x^{3} - 8\,n - 16\,n x - 9\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(8 + 24\,x + 25\,x^{2} + 10\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
376114
417239150
5417648121178
69923320482151634
72396827654281251812258
85771935264919077751772118042
913935368868076564010127223824128642
1033631464127312285481835875061946308271102010
11811939406833131007447693038652129082824387301363314
1219601104935247985340661301510180098739202081039334753311293962
1347321276996723624111336411213557772883794981312723188753005723614608514
141142437258492077164353899839982453157539179030273677221978946895695267839116413466
152758071890258588079710992148137702071221196678949293774616133950935076666671891679342158457426
Technical details
Node id
n-a1892d0a9ea5b20e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b42d277a4b716fed

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:    7,     6,     1;
   4:   17,    23,     9,     1;
   5:   41,    76,    48,    12,     1;
   6:   99,   233,   204,    82,    15,    1;
   7:  239,   682,   765,   428,   125,   18,    1;
   8:  577,  1935,  2649,  1907,   775,  177,   21,   1;
   9: 1393,  5368,  8680,  7656,  4010, 1272,  238,  24,  1;
  10: 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (2*n^2-9*n+10+(2*n^2-12*n+16)*x-(2*n-4)*x^2)*P(n-1,x) - (8*n-28+(16*n-59)*x+(9*n-36)*x^2+(n-5)*x^3)*d/dx P(n-1,x) + (8+24*x+25*x^2+10*x^3+x^4)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 9*n + 10 + (2*n^2 - 12*n + 16)*x - (2*n - 4)*x^2)*P[n - 1, x] - (8*n - 28 + (16*n - 59)*x + (9*n - 36)*x^2 + (n - 5)*x^3)*D[P[n - 1, x], x] + (8 + 24*x + 25*x^2 + 10*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = ((2*n^2 - 9*n + 10 + (2*n^2 - 12*n + 16)*x - (2*n - 4)*x^2)*v[i - 1] - (8*n - 28 + (16*n - 59)*x + (9*n - 36)*x^2 + (n - 5)*x^3)*deriv(v[i - 1], x) + (8 + 24*x + 25*x^2 + 10*x^3 + x^4)*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, 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