Real-rooted atlas

A105868 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(7\,n x - 7\,n x^{2} + n x^{3} + n^{2} - 6\,n^{2} x + 2\,n^{2} x^{2} - n^{3} + n^{3} x\right)\,P_{n}(x) = \left(24\,x - 32\,x^{2} + 16\,x^{3} - 2\,x^{4} + 12\,n - 48\,n x + 52\,n x^{2} - 26\,n x^{3} + 3\,n x^{4} - 16\,n^{2} + 33\,n^{2} x - 26\,n^{2} x^{2} + 6\,n^{2} x^{3} + 4\,n^{3} - 7\,n^{3} x + 3\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(16\,x^{2} - 44\,x^{3} + 18\,x^{4} - 2\,x^{5} + 8\,n x - 38\,n x^{2} + 25\,n x^{3} - 4\,n x^{4} - 8\,n^{2} x + 10\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
22417
30129122
40124216171
50090110251226
60060380240361717
700056011904624912262
8000280287030808126417107
900003150107106972133281122246
10000012601915232550142802070100169413
11000001663283622854702706030801211215986
120000055441173482938322009704818044221441670441
130000008408458858888288243329081510616216912076686
140000002402467610423543522354352870870132132837219616420403
Technical details
Node id
n-4f83e0963e8ad59c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d8528ff58087f621

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: 1,  1;
  2: 2,  4,  1;
  3: 0, 12,  9,   1;
  4: 0, 12, 42,  16,    1;
  5: 0,  0, 90, 110,   25,     1;
  6: 0,  0, 60, 380,  240,    36,    1;
  7: 0,  0,  0, 560, 1190,   462,   49,    1;
  8: 0,  0,  0, 280, 2870,  3080,  812,   64,  1;
  9: 0,  0,  0,   0, 3150, 10710, 6972, 1332, 81, 1;
%F Formula
The row polynomials satisfy -(n^3-n^2-(n^3-6*n^2+7*n)*x-(2*n^2-7*n)*x^2-n*x^3)*P(n,x) = (4*n^3-16*n^2+12*n-(7*n^3-33*n^2+48*n-24)*x+(3*n^3-26*n^2+52*n-32)*x^2+(6*n^2-26*n+16)*x^3+(3*n-2)*x^4)*P(n-1,x) - ((8*n^2-8*n)*x-(10*n^2-38*n+16)*x^2+(2*n^2-25*n+44)*x^3+(4*n-18)*x^4+2*x^5)*d/dx 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[((4*n^3 - 16*n^2 + 12*n - (7*n^3 - 33*n^2 + 48*n - 24)*x + (3*n^3 - 26*n^2 + 52*n - 32)*x^2 + (6*n^2 - 26*n + 16)*x^3 + (3*n - 2)*x^4)*P[n - 1, x] - ((8*n^2 - 8*n)*x - (10*n^2 - 38*n + 16)*x^2 + (2*n^2 - 25*n + 44)*x^3 + (4*n - 18)*x^4 + 2*x^5)*D[P[n - 1, x], x])/(-n^3 + n^2 + (n^3 - 6*n^2 + 7*n)*x + (2*n^2 - 7*n)*x^2 + n*x^3)]];
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] = ((4*n^3 - 16*n^2 + 12*n - (7*n^3 - 33*n^2 + 48*n - 24)*x + (3*n^3 - 26*n^2 + 52*n - 32)*x^2 + (6*n^2 - 26*n + 16)*x^3 + (3*n - 2)*x^4)*v[i - 1] - ((8*n^2 - 8*n)*x - (10*n^2 - 38*n + 16)*x^2 + (2*n^2 - 25*n + 44)*x^3 + (4*n - 18)*x^4 + 2*x^5)*deriv(v[i - 1], x))/(-n^3 + n^2 + (n^3 - 6*n^2 + 7*n)*x + (2*n^2 - 7*n)*x^2 + n*x^3)); 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

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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.