Real-rooted atlas

A114656 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(1 - x - 11\,x^{2} - 15\,x^{3} - 6\,x^{4} + 4\,n + 17\,n x + 25\,n x^{2} + 15\,n x^{3} + 3\,n x^{4}\right)\,P_{n-1}(x) + \left(1 + 5\,x + 6\,x^{2} - 2\,x^{3} - 7\,x^{4} - 3\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2358
311383180
445237399215896
5197139635903972159710752
6903804527910471303875512425135168
7427945930200805457660573585374970999551757184
8207932611491378293396210567001556666695361525582467523429120

All 15 rows

Technical details
Node id
n-819d2b81c1d2dd75 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-fb2c44332fe49020

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      3,       5;
   3:     11,      38,       31;
   4:     45,     237,      399,       215;
   5:    197,    1396,     3590,      3972,      1597;
   6:    903,    8045,    27910,     47130,     38755,      12425;
   7:   4279,   45930,   200805,    457660,    573585,     374970,      99955;
   8:  20793,  261149,  1378293,   3962105,   6700155,    6666695,    3615255,    824675;
   9: 103049, 1482440,  9169804,  31850840,  67947334,   91171640,   75158188,  34812968,   6939769;
  10: 518859, 8412021, 59681484, 243172356, 627053994, 1061268390, 1179042396, 829298196, 335188323, 59334605;
%F Formula
The row polynomials satisfy (n+1+(2*n+2)*x+(n+1)*x^2)*P(n,x) = (4*n+1+(17*n-1)*x+(25*n-11)*x^2+(15*n-15)*x^3+(3*n-6)*x^4)*P(n-1,x) + (1+5*x+6*x^2-2*x^3-7*x^4-3*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 1 + (17*n - 1)*x + (25*n - 11)*x^2 + (15*n - 15)*x^3 + (3*n - 6)*x^4)*P[n - 1, x] + (1 + 5*x + 6*x^2 - 2*x^3 - 7*x^4 - 3*x^5)*D[P[n - 1, x], x])/(n + 1 + (2*n + 2)*x + (n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n + 1 + (17*n - 1)*x + (25*n - 11)*x^2 + (15*n - 15)*x^3 + (3*n - 6)*x^4)*v[i - 1] + (1 + 5*x + 6*x^2 - 2*x^3 - 7*x^4 - 3*x^5)*deriv(v[i - 1], x))/(n + 1 + (2*n + 2)*x + (n + 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023