Real-rooted atlas

A046716 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 4\,n\right)\,P_{n}(x) = \left(132 + 864\,x + 1872\,x^{2} + 1344\,x^{3} - 116\,n - 792\,n x - 1776\,n x^{2} - 1312\,n x^{3} + 36\,n^{2} + 240\,n^{2} x + 528\,n^{2} x^{2} + 384\,n^{2} x^{3} - 4\,n^{3} - 24\,n^{3} x - 48\,n^{3} x^{2} - 32\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(24 + 204\,x + 648\,x^{2} + 912\,x^{3} + 480\,x^{4} - 23\,n - 200\,n x - 648\,n x^{2} - 928\,n x^{3} - 496\,n x^{4} + 8\,n^{2} + 68\,n^{2} x + 216\,n^{2} x^{2} + 304\,n^{2} x^{3} + 160\,n^{2} x^{4} - n^{3} - 8\,n^{3} x - 24\,n^{3} x^{2} - 32\,n^{3} x^{3} - 16\,n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-60 - 224\,x - 192\,x^{2} + 52\,n + 208\,n x + 192\,n x^{2} - 12\,n^{2} - 48\,n^{2} x - 48\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-8 - 68\,x - 184\,x^{2} - 160\,x^{3} + 13\,n + 96\,n x + 236\,n x^{2} + 192\,n x^{3} - 6\,n^{2} - 42\,n^{2} x - 96\,n^{2} x^{2} - 72\,n^{2} x^{3} + n^{3} + 6\,n^{3} x + 12\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(20 + 64\,x - 24\,n - 64\,n x + 8\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-8 - 36\,x - 40\,x^{2} + 5\,n + 24\,n x + 28\,n x^{2} - n^{2} - 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)
\(P_{3}(x) = 10 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
222
310818
4508824162
5288856672641880
61922843212264601616028794
7146268675219130418585667360384546282
812505094504828367044224576311936091353689612165170

All 15 rows

Technical details
Node id
n-4d1bc17697375f12 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-574213cee5e097d9

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:        0;
   2:        2;
   3:       10,         8;
   4:       50,        88,        24;
   5:      288,       856,       672,         64;
   6:     1922,      8432,     12264,       6016,        160;
   7:    14626,     86752,    191304,     185856,      67360,        384;
   8:   125050,    945048,   2836704,    4224576,    3119360,     913536,        896;
   9:  1187472,  10971120,  42000192,   85236416,   96661760,   58041600,   14407680,      2048;
  10: 12408738, 136019808, 636161424, 1645056384, 2539208160, 2338438656, 1189090560, 257396736, 4608;
%F Formula
The row polynomials satisfy (4*n-8)*P(n,x) = -(4*n^3-36*n^2+116*n-132+(24*n^3-240*n^2+792*n-864)*x+(48*n^3-528*n^2+1776*n-1872)*x^2+(32*n^3-384*n^2+1312*n-1344)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(8*n^3-68*n^2+200*n-204)*x+(24*n^3-216*n^2+648*n-648)*x^2+(32*n^3-304*n^2+928*n-912)*x^3+(16*n^3-160*n^2+496*n-480)*x^4)*d/dx P(n-3,x) - (12*n^2-52*n+60+(48*n^2-208*n+224)*x+(48*n^2-192*n+192)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(6*n^3-42*n^2+96*n-68)*x+(12*n^3-96*n^2+236*n-184)*x^2+(8*n^3-72*n^2+192*n-160)*x^3)*d/dx P(n-2,x) + (8*n^2-24*n+20+(16*n^2-64*n+64)*x)*P(n-1,x) - (n^2-5*n+8+(4*n^2-24*n+36)*x+(4*n^2-28*n+40)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 2, P(3,x) = 10 + 8*x.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[3, x_] := 10 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 36*n^2 + 116*n - 132 + (24*n^3 - 240*n^2 + 792*n - 864)*x + (48*n^3 - 528*n^2 + 1776*n - 1872)*x^2 + (32*n^3 - 384*n^2 + 1312*n - 1344)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*D[P[n - 3, x], x] - (12*n^2 - 52*n + 60 + (48*n^2 - 208*n + 224)*x + (48*n^2 - 192*n + 192)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 24*n + 20 + (16*n^2 - 64*n + 64)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*D[P[n - 1, x], x])/(4*n - 8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; v[3] = 10 + 8*x; for(n = 4, N, my(i = n); v[i] = (-(4*n^3 - 36*n^2 + 116*n - 132 + (24*n^3 - 240*n^2 + 792*n - 864)*x + (48*n^3 - 528*n^2 + 1776*n - 1872)*x^2 + (32*n^3 - 384*n^2 + 1312*n - 1344)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*deriv(v[i - 3], x) - (12*n^2 - 52*n + 60 + (48*n^2 - 208*n + 224)*x + (48*n^2 - 192*n + 192)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 24*n + 20 + (16*n^2 - 64*n + 64)*x)*v[i - 1] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*deriv(v[i - 1], x))/(4*n - 8)); 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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023