Real-rooted atlas

A091044 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(n - 13\,n x + 6\,n x^{2} - 5\,n^{2} + 12\,n^{2} x + 6\,n^{3}\right)\,P_{n}(x) = \left(21\,x^{2} - 18\,x^{3} - 8\,n - 73\,n x - 75\,n x^{2} + 54\,n x^{3} + 4\,n^{2} + 21\,n^{2} x + 108\,n^{2} x^{2} + 24\,n^{3} + 54\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x^{2} - 6\,x^{3} - 36\,x^{4} - 8\,n x - 48\,n x^{2} - 72\,n x^{3} - 12\,n^{2} x - 36\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110616
26410227193
335210086841082152
4179275849504378040523065
58704491529331269552186301458240808
6409602903047488008605444354568505051032466217
7188416160972852807688363520660960024494403674161749624886384

All 15 rows

Technical details
Node id
n-2f1b46ca121c31c9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8c52ef5185c6c457

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:      10,        6;
  2:      64,      102,        27;
  3:     352,     1008,       684,       108;
  4:    1792,     7584,      9504,      3780,       405;
  5:    8704,    49152,     93312,     69552,     18630,      1458;
  6:   40960,   290304,    748800,    860544,    435456,     85050,      5103;
  7:  188416,  1609728,   5280768,   8363520,   6609600,   2449440,    367416,    17496;
  8:  851968,  8527872,  34062336,  69368832,  76412160,  44649792,  12737088,  1522152,   59049;
  9: 3801088, 43646976, 205848576, 514805760, 735920640, 604910592, 274057344, 62355744, 6101730, 196830;
%F Formula
The row polynomials satisfy (6*n^3-5*n^2+n+(12*n^2-13*n)*x+6*n*x^2)*P(n,x) = (24*n^3+4*n^2-8*n+(54*n^3+21*n^2-73*n)*x+(108*n^2-75*n+21)*x^2+(54*n-18)*x^3)*P(n-1,x) - ((12*n^2+8*n)*x+(36*n^2+48*n-2)*x^2+(72*n+6)*x^3+36*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[((24*n^3 + 4*n^2 - 8*n + (54*n^3 + 21*n^2 - 73*n)*x + (108*n^2 - 75*n + 21)*x^2 + (54*n - 18)*x^3)*P[n - 1, x] - ((12*n^2 + 8*n)*x + (36*n^2 + 48*n - 2)*x^2 + (72*n + 6)*x^3 + 36*x^4)*D[P[n - 1, x], x])/(6*n^3 - 5*n^2 + n + (12*n^2 - 13*n)*x + 6*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] = ((24*n^3 + 4*n^2 - 8*n + (54*n^3 + 21*n^2 - 73*n)*x + (108*n^2 - 75*n + 21)*x^2 + (54*n - 18)*x^3)*v[i - 1] - ((12*n^2 + 8*n)*x + (36*n^2 + 48*n - 2)*x^2 + (72*n + 6)*x^3 + 36*x^4)*deriv(v[i - 1], x))/(6*n^3 - 5*n^2 + n + (12*n^2 - 13*n)*x + 6*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