Real-rooted atlas

A172431 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(10 - 17\,x + 4\,x^{2} - 42\,n + 59\,n x - 12\,n x^{2} + 60\,n^{2} - 58\,n^{2} x + 8\,n^{2} x^{2} - 36\,n^{3} + 16\,n^{3} x + 8\,n^{4}\right)\,P_{n}(x) = \left(2 + 7\,x + 13\,x^{2} - 4\,x^{3} - 10\,n - 26\,n x + 79\,n x^{2} - 28\,n x^{3} + 20\,n^{2} + 52\,n^{2} x - 126\,n^{2} x^{2} + 24\,n^{2} x^{3} - 20\,n^{3} - 68\,n^{3} x + 48\,n^{3} x^{2} + 8\,n^{4} + 24\,n^{4} x\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} - 4\,n x - 7\,n x^{2} + 32\,n x^{3} - 16\,n x^{4} + 6\,n^{2} x + 16\,n^{2} x^{2} - 32\,n^{2} x^{3} - 4\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
210616
326441282
45016212020352
582408588260301368
612283018721624490424980
717014764620636037808405617302

All 15 rows

Technical details
Node id
n-cf0fcb1a37e01332 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3ef6031d1e6db40f

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 -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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   2;
  2:  10,    6;
  3:  26,   44,    12;
  4:  50,  162,   120,    20;
  5:  82,  408,   588,   260,    30;
  6: 122,  830,  1872,  1624,   490,    42;
  7: 170, 1476,  4620,  6360,  3780,   840,    56;
  8: 226, 2394,  9672, 18700, 17820,  7812,  1344,   72;
  9: 290, 3632, 18060, 45500, 61490, 43560, 14784, 2040, 90;
%F Formula
The row polynomials satisfy (8*n^4-36*n^3+60*n^2-42*n+10+(16*n^3-58*n^2+59*n-17)*x+(8*n^2-12*n+4)*x^2)*P(n,x) = (8*n^4-20*n^3+20*n^2-10*n+2+(24*n^4-68*n^3+52*n^2-26*n+7)*x+(48*n^3-126*n^2+79*n+13)*x^2+(24*n^2-28*n-4)*x^3)*P(n-1,x) - ((4*n^3-6*n^2+4*n-1)*x+(16*n^3-16*n^2+7*n-4)*x^2+(32*n^2-32*n)*x^3+16*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^4 - 20*n^3 + 20*n^2 - 10*n + 2 + (24*n^4 - 68*n^3 + 52*n^2 - 26*n + 7)*x + (48*n^3 - 126*n^2 + 79*n + 13)*x^2 + (24*n^2 - 28*n - 4)*x^3)*P[n - 1, x] - ((4*n^3 - 6*n^2 + 4*n - 1)*x + (16*n^3 - 16*n^2 + 7*n - 4)*x^2 + (32*n^2 - 32*n)*x^3 + 16*n*x^4)*D[P[n - 1, x], x])/(8*n^4 - 36*n^3 + 60*n^2 - 42*n + 10 + (16*n^3 - 58*n^2 + 59*n - 17)*x + (8*n^2 - 12*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^4 - 20*n^3 + 20*n^2 - 10*n + 2 + (24*n^4 - 68*n^3 + 52*n^2 - 26*n + 7)*x + (48*n^3 - 126*n^2 + 79*n + 13)*x^2 + (24*n^2 - 28*n - 4)*x^3)*v[i - 1] - ((4*n^3 - 6*n^2 + 4*n - 1)*x + (16*n^3 - 16*n^2 + 7*n - 4)*x^2 + (32*n^2 - 32*n)*x^3 + 16*n*x^4)*deriv(v[i - 1], x))/(8*n^4 - 36*n^3 + 60*n^2 - 42*n + 10 + (16*n^3 - 58*n^2 + 59*n - 17)*x + (8*n^2 - 12*n + 4)*x^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

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