Real-rooted atlas

A085478 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-48 + 63\,x - 12\,x^{2} + 80\,n - 66\,n x + 8\,n x^{2} - 44\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(-24 - 42\,x + 175\,x^{2} - 44\,x^{3} + 52\,n + 136\,n x - 182\,n x^{2} + 24\,n x^{3} - 36\,n^{2} - 108\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(-6\,x - 11\,x^{2} + 48\,x^{3} - 16\,x^{4} + 10\,n x + 32\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
345110
46179133
583745141105
6106513395201324
712101297364175271977
81414556110058402943512895

All 15 rows

Technical details
Node id
n-26e4387737b6859f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-285a8f351a67f82b

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:  2,   1;
   3:  4,   5,    1;
   4:  6,  17,    9,    1;
   5:  8,  37,   45,   14,    1;
   6: 10,  65,  133,   95,   20,    1;
   7: 12, 101,  297,  364,  175,   27,    1;
   8: 14, 145,  561, 1005,  840,  294,   35,   1;
   9: 16, 197,  949, 2266, 2805, 1722,  462,  44,  1;
  10: 18, 257, 1485, 4459, 7436, 6831, 3234, 690, 54, 1;
%F Formula
The row polynomials satisfy (8*n^3-44*n^2+80*n-48+(16*n^2-66*n+63)*x+(8*n-12)*x^2)*P(n,x) = (8*n^3-36*n^2+52*n-24+(24*n^3-108*n^2+136*n-42)*x+(48*n^2-182*n+175)*x^2+(24*n-44)*x^3)*P(n-1,x) - ((4*n^2-10*n+6)*x+(16*n^2-32*n+11)*x^2+(32*n-48)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 36*n^2 + 52*n - 24 + (24*n^3 - 108*n^2 + 136*n - 42)*x + (48*n^2 - 182*n + 175)*x^2 + (24*n - 44)*x^3)*P[n - 1, x] - ((4*n^2 - 10*n + 6)*x + (16*n^2 - 32*n + 11)*x^2 + (32*n - 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 44*n^2 + 80*n - 48 + (16*n^2 - 66*n + 63)*x + (8*n - 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 36*n^2 + 52*n - 24 + (24*n^3 - 108*n^2 + 136*n - 42)*x + (48*n^2 - 182*n + 175)*x^2 + (24*n - 44)*x^3)*v[i - 1] - ((4*n^2 - 10*n + 6)*x + (16*n^2 - 32*n + 11)*x^2 + (32*n - 48)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 44*n^2 + 80*n - 48 + (16*n^2 - 66*n + 63)*x + (8*n - 12)*x^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. [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