Real-rooted atlas

A125662 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 8\,n x - 28\,n x^{2} + 10\,n x^{3} - 14\,n^{2} - 6\,n^{2} x + 20\,n^{2} x^{2} + 12\,n^{3} + 10\,n^{3} x\right)\,P_{n}(x) = \left(19\,x^{2} + 37\,x^{3} + 5\,x^{4} - 25\,x^{5} - 2\,n - 38\,n x - 111\,n x^{2} - 121\,n x^{3} - 25\,n x^{4} + 25\,n x^{5} - 2\,n^{2} - 36\,n^{2} x - 27\,n^{2} x^{2} + 75\,n^{2} x^{3} + 50\,n^{2} x^{4} + 12\,n^{3} + 64\,n^{3} x + 75\,n^{3} x^{2} + 25\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(2\,x^{2} + 11\,x^{3} - x^{4} - 35\,x^{5} - 25\,x^{6} - 2\,n x - 19\,n x^{2} - 62\,n x^{3} - 95\,n x^{4} - 50\,n x^{5} - 6\,n^{2} x - 41\,n^{2} x^{2} - 60\,n^{2} x^{3} - 25\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
2722837
310598821178
41311432131055813
516187788143410201443589
619278157042845721321637715465
7223872748100752017021207984898765444

All 15 rows

Technical details
Node id
n-edade370562732e6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-70a162a802c85a2a

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,  22,    8;
  3: 10,  59,   88,    21;
  4: 13, 114,  321,   310,     55;
  5: 16, 187,  788,  1434,   1020,    144;
  6: 19, 278, 1570,  4284,   5721,   3216,    377;
  7: 22, 387, 2748, 10075,  20170,  21207,   9848,    987;
  8: 25, 514, 4403, 20346,  54780,  86250,  74657,  29522,  2584;
  9: 28, 659, 6616, 36960, 125580, 265929, 344212, 252958, 87080, 6765;
%F Formula
The row polynomials satisfy (12*n^3-14*n^2+4*n+(10*n^3-6*n^2-8*n)*x+(20*n^2-28*n)*x^2+10*n*x^3)*P(n,x) = (12*n^3-2*n^2-2*n+(64*n^3-36*n^2-38*n)*x+(75*n^3-27*n^2-111*n+19)*x^2+(25*n^3+75*n^2-121*n+37)*x^3+(50*n^2-25*n+5)*x^4+(25*n-25)*x^5)*P(n-1,x) - ((6*n^2+2*n)*x+(41*n^2+19*n-2)*x^2+(60*n^2+62*n-11)*x^3+(25*n^2+95*n+1)*x^4+(50*n+35)*x^5+25*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 + (64*n^3 - 36*n^2 - 38*n)*x + (75*n^3 - 27*n^2 - 111*n + 19)*x^2 + (25*n^3 + 75*n^2 - 121*n + 37)*x^3 + (50*n^2 - 25*n + 5)*x^4 + (25*n - 25)*x^5)*P[n - 1, x] - ((6*n^2 + 2*n)*x + (41*n^2 + 19*n - 2)*x^2 + (60*n^2 + 62*n - 11)*x^3 + (25*n^2 + 95*n + 1)*x^4 + (50*n + 35)*x^5 + 25*x^6)*D[P[n - 1, x], x])/(12*n^3 - 14*n^2 + 4*n + (10*n^3 - 6*n^2 - 8*n)*x + (20*n^2 - 28*n)*x^2 + 10*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 + (64*n^3 - 36*n^2 - 38*n)*x + (75*n^3 - 27*n^2 - 111*n + 19)*x^2 + (25*n^3 + 75*n^2 - 121*n + 37)*x^3 + (50*n^2 - 25*n + 5)*x^4 + (25*n - 25)*x^5)*v[i - 1] - ((6*n^2 + 2*n)*x + (41*n^2 + 19*n - 2)*x^2 + (60*n^2 + 62*n - 11)*x^3 + (25*n^2 + 95*n + 1)*x^4 + (50*n + 35)*x^5 + 25*x^6)*deriv(v[i - 1], x))/(12*n^3 - 14*n^2 + 4*n + (10*n^3 - 6*n^2 - 8*n)*x + (20*n^2 - 28*n)*x^2 + 10*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023