Real-rooted atlas

A110441 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 24\,n x + 4\,n x^{2} + 2\,n x^{3} - 14\,n^{2} + 18\,n^{2} x + 4\,n^{2} x^{2} + 12\,n^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(23\,x^{2} - 7\,x^{3} - 7\,x^{4} - x^{5} - 2\,n - 42\,n x - 99\,n x^{2} + 11\,n x^{3} + 11\,n x^{4} + n x^{5} - 2\,n^{2} - 24\,n^{2} x + 81\,n^{2} x^{2} + 27\,n^{2} x^{3} + 2\,n^{2} x^{4} + 12\,n^{3} + 56\,n^{3} x + 15\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(2\,x^{2} + 7\,x^{3} - 29\,x^{4} - 11\,x^{5} - x^{6} - 2\,n x - 23\,n x^{2} - 70\,n x^{3} - 23\,n x^{4} - 2\,n x^{5} - 6\,n^{2} x - 37\,n^{2} x^{2} - 12\,n^{2} x^{3} - n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
2720734
310556815148
41310826419631612
5161796761002516632452
61926813853216330912841279608
72237524727945129109969307625537024

All 15 rows

Technical details
Node id
n-3c9d2e7be059a416 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1f9cbdd862ac2135

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:  4,   3;
  2:  7,  20,    7;
  3: 10,  55,   68,    15;
  4: 13, 108,  264,   196,    31;
  5: 16, 179,  676,  1002,   516,     63;
  6: 19, 268, 1385,  3216,  3309,   1284,    127;
  7: 22, 375, 2472,  7945, 12910,   9969,   3076,   255;
  8: 25, 500, 4018, 16620, 37680,  46140,  28150,  7172,   511;
  9: 28, 643, 6104, 30996, 91056, 156414, 151544, 75772, 16388, 1023;
%F Formula
The row polynomials satisfy (12*n^3-14*n^2+4*n+(2*n^3+18*n^2-24*n)*x+(4*n^2+4*n)*x^2+2*n*x^3)*P(n,x) = (12*n^3-2*n^2-2*n+(56*n^3-24*n^2-42*n)*x+(15*n^3+81*n^2-99*n+23)*x^2+(n^3+27*n^2+11*n-7)*x^3+(2*n^2+11*n-7)*x^4+(n-1)*x^5)*P(n-1,x) - ((6*n^2+2*n)*x+(37*n^2+23*n-2)*x^2+(12*n^2+70*n-7)*x^3+(n^2+23*n+29)*x^4+(2*n+11)*x^5+x^6)*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[((12*n^3 - 2*n^2 - 2*n + (56*n^3 - 24*n^2 - 42*n)*x + (15*n^3 + 81*n^2 - 99*n + 23)*x^2 + (n^3 + 27*n^2 + 11*n - 7)*x^3 + (2*n^2 + 11*n - 7)*x^4 + (n - 1)*x^5)*P[n - 1, x] - ((6*n^2 + 2*n)*x + (37*n^2 + 23*n - 2)*x^2 + (12*n^2 + 70*n - 7)*x^3 + (n^2 + 23*n + 29)*x^4 + (2*n + 11)*x^5 + x^6)*D[P[n - 1, x], x])/(12*n^3 - 14*n^2 + 4*n + (2*n^3 + 18*n^2 - 24*n)*x + (4*n^2 + 4*n)*x^2 + 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] = ((12*n^3 - 2*n^2 - 2*n + (56*n^3 - 24*n^2 - 42*n)*x + (15*n^3 + 81*n^2 - 99*n + 23)*x^2 + (n^3 + 27*n^2 + 11*n - 7)*x^3 + (2*n^2 + 11*n - 7)*x^4 + (n - 1)*x^5)*v[i - 1] - ((6*n^2 + 2*n)*x + (37*n^2 + 23*n - 2)*x^2 + (12*n^2 + 70*n - 7)*x^3 + (n^2 + 23*n + 29)*x^4 + (2*n + 11)*x^5 + x^6)*deriv(v[i - 1], x))/(12*n^3 - 14*n^2 + 4*n + (2*n^3 + 18*n^2 - 24*n)*x + (4*n^2 + 4*n)*x^2 + 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. [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