Real-rooted atlas

A125662 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n - n x - n x^{2} + n^{3}\right)\,P_{n}(x) = \left(3 - 3\,x - 4\,x^{2} - x^{3} + 7\,n - n x - 4\,n x^{2} - n x^{3} + 6\,n^{2} + 3\,n^{2} x + 2\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-3 - 13\,x - 16\,x^{2} - 7\,x^{3} - x^{4} + 3\,n + 4\,n x + n x^{2} + 3\,n^{2} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
23124459
3148188728416
46731208756192162845
529626918611225284803219032
6127353675242264246407600115264125207
75380018520826337619976086720213122688128812992

All 15 rows

Technical details
Node id
n-5d602378ea445bcb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-87f68aa23f424d68

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,       2;
  2:     31,      24,       4;
  3:    148,     188,      72,       8;
  4:    673,    1208,     756,     192,      16;
  5:   2962,    6918,    6112,    2528,     480,      32;
  6:  12735,   36752,   42264,   24640,    7600,    1152,     64;
  7:  53800,  185208,  263376,  199760,   86720,   21312,   2688,    128;
  8: 224193,  897552, 1523928, 1429312,  806640,  278656,  56896,   6144,   256;
  9: 924062, 4220682, 8341344, 9343328, 6518624, 2925216, 838656, 146432, 13824, 512;
%F Formula
The row polynomials satisfy (n^3+2*n-n*x-n*x^2)*P(n,x) = (2*n^3+6*n^2+7*n+3+(n^3+3*n^2-n-3)*x-(4*n+4)*x^2-(n+1)*x^3)*P(n-1,x) + (3*n^2+3*n-3+(4*n^2+4*n-13)*x+(n^2+n-16)*x^2-7*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[((2*n^3 + 6*n^2 + 7*n + 3 + (n^3 + 3*n^2 - n - 3)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (3*n^2 + 3*n - 3 + (4*n^2 + 4*n - 13)*x + (n^2 + n - 16)*x^2 - 7*x^3 - x^4)*D[P[n - 1, x], x])/(n^3 + 2*n - n*x - 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] = ((2*n^3 + 6*n^2 + 7*n + 3 + (n^3 + 3*n^2 - n - 3)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*v[i - 1] + (3*n^2 + 3*n - 3 + (4*n^2 + 4*n - 13)*x + (n^2 + n - 16)*x^2 - 7*x^3 - x^4)*deriv(v[i - 1], x))/(n^3 + 2*n - n*x - 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