Real-rooted atlas

A123519 → reverse → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(6 + 27\,x + 42\,x^{2} + 27\,x^{3} + 6\,x^{4} + 4\,n + 20\,n x + 34\,n x^{2} + 24\,n x^{3} + 6\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 12\,x^{2} - 24\,x^{3} - 20\,x^{4} - 6\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 4 + 14\,x + 11\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04141129
18447641169
216120324372153985
3323041120199617185715741
46473634408352110967642213133461
512817289792301605448057672331127953195025
625639682643298752226160324912285816140712296811136689
75128960686083015048375361524304181680013670165890981107716625109

All 15 rows

Technical details
Node id
n-9203eb9e5a9cd0d4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-65206dde5faab998

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:    4,    14,     11;
  1:    8,    44,     76,      41;
  2:   16,   120,    324,     372,     153;
  3:   32,   304,   1120,    1996,    1718,      571;
  4:   64,   736,   3440,    8352,   11096,     7642,     2131;
  5:  128,  1728,   9792,   30160,   54480,    57672,    33112,     7953;
  6:  256,  3968,  26432,   98752,  226160,   324912,   285816,   140712,    29681;
  7:  512,  8960,  68608,  301504,  837536,  1524304,  1816800,  1367016,   589098,   110771;
  8: 1024, 19968, 172800,  873472, 2854656,  6300000,  9504656,  9676320,  6360444,  2437222,  413403;
  9: 2048, 44032, 424960, 2429184, 9134592, 23717120, 43371776, 55846960, 49609368, 28949204, 9986596, 1542841;
%F Formula
The row polynomials satisfy (2*n+3+(4*n+6)*x+(2*n+3)*x^2)*P(n,x) = (4*n+6+(20*n+27)*x+(34*n+42)*x^2+(24*n+27)*x^3+(6*n+6)*x^4)*P(n-1,x) - (2*x+12*x^2+24*x^3+20*x^4+6*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 14*x + 11*x^2.
%t Mathematica
P[0, x_] := 4 + 14*x + 11*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 6 + (20*n + 27)*x + (34*n + 42)*x^2 + (24*n + 27)*x^3 + (6*n + 6)*x^4)*P[n - 1, x] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*D[P[n - 1, x], x])/(2*n + 3 + (4*n + 6)*x + (2*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 4 + 14*x + 11*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 6 + (20*n + 27)*x + (34*n + 42)*x^2 + (24*n + 27)*x^3 + (6*n + 6)*x^4)*v[i - 1] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*deriv(v[i - 1], x))/(2*n + 3 + (4*n + 6)*x + (2*n + 3)*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

Candidate equalities ?

Same fingerprint ?

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023