Real-rooted atlas

A207824 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-14\,n - 14\,n x - 2\,n x^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-51 - 51\,x - 13\,x^{2} - x^{3} - 41\,n - 49\,n x - 13\,n x^{2} - n x^{3} + 15\,n^{2} + 3\,n^{2} x + 5\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-336 - 349\,x - 127\,x^{2} - 19\,x^{3} - x^{4} + 21\,n + 10\,n x + n x^{2} + 21\,n^{2} + 10\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1617
23412147
3188103181310
410217642072412017
554685208192834630112981
6289593357215831388052036182799
7151976207921119430372856820729421524204

All 15 rows

Technical details
Node id
n-fb9bfbbcf93b0a63 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-de33d2c8f30482a7

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:       6,       1;
  2:      34,      12,       1;
  3:     188,     103,      18,       1;
  4:    1021,     764,     207,      24,      1;
  5:    5468,    5208,    1928,     346,     30,      1;
  6:   28959,   33572,   15831,    3880,    520,     36,     1;
  7:  151976,  207921,  119430,   37285,   6820,    729,    42,    1;
  8:  791526, 1249380,  847515,  322360,  75090,  10948,   973,   48,  1;
  9: 4096030, 7331655, 5741640, 2579830, 732130, 135891, 16464, 1252, 54, 1;
%F Formula
The row polynomials satisfy (2*n^3-14*n-14*n*x-2*n*x^2)*P(n,x) = (5*n^3+15*n^2-41*n-51+(n^3+3*n^2-49*n-51)*x-(13*n+13)*x^2-(n+1)*x^3)*P(n-1,x) + (21*n^2+21*n-336+(10*n^2+10*n-349)*x+(n^2+n-127)*x^2-19*x^3-x^4)*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[((5*n^3 + 15*n^2 - 41*n - 51 + (n^3 + 3*n^2 - 49*n - 51)*x - (13*n + 13)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (21*n^2 + 21*n - 336 + (10*n^2 + 10*n - 349)*x + (n^2 + n - 127)*x^2 - 19*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 14*n - 14*n*x - 2*n*x^2)]];
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] = ((5*n^3 + 15*n^2 - 41*n - 51 + (n^3 + 3*n^2 - 49*n - 51)*x - (13*n + 13)*x^2 - (n + 1)*x^3)*v[i - 1] + (21*n^2 + 21*n - 336 + (10*n^2 + 10*n - 349)*x + (n^2 + n - 127)*x^2 - 19*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 14*n - 14*n*x - 2*n*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