Real-rooted atlas

A111125 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0) · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(-3 - 6\,x - 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-1 + 4\,x + 21\,x^{2} + 26\,x^{3} + 10\,x^{4} + 2\,n + 3\,n x - 5\,n x^{2} - 11\,n x^{3} - 5\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 17\,x^{2} + 27\,x^{3} + 19\,x^{4} + 5\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2347
35151131
47355629127
596617118976511
6111104077156051992047
7131698322054273018725218191
81524515334965918098105655136432767
917340261810642255853777433745167793571131071
101945642182086262453119168146357112309490969349524287
1121595648938160138005326109515340541646364245142065244762097151
12237599614660332820958012971561861210512119333571157015407353640798388607
132595013805109175541475180860042220506997390821920067019003612900115920016776133554431
142711701930517374598670038101051042335020630790297251103093255022676760111213003277611439204134217727
152914212639026767017210057579585238962905528154094906560120902740112915270751859803382006192165191149851536870911

Generated b-files: A083420 (row sums), A200073 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-ac316f680714c5b7 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-347434f4a19cf0c5

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  3,   4;
   3:  5,  15,   11;
   4:  7,  35,   56,    29;
   5:  9,  66,  171,   189,    76;
   6: 11, 110,  407,   715,   605,    199;
   7: 13, 169,  832,  2054,  2730,   1872,    521;
   8: 15, 245, 1533,  4965,  9180,   9810,   5655,   1364;
   9: 17, 340, 2618, 10642, 25585,  37774,  33745,  16779,  3571;
  10: 19, 456, 4218, 20862, 62453, 119168, 146357, 112309, 49096, 9349;
%F Formula
The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = (2*n-1+(3*n+4)*x-(5*n-21)*x^2-(11*n-26)*x^3-(5*n-10)*x^4)*P(n-1,x) + (4*x+17*x^2+27*x^3+19*x^4+5*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[((2*n - 1 + (3*n + 4)*x - (5*n - 21)*x^2 - (11*n - 26)*x^3 - (5*n - 10)*x^4)*P[n - 1, x] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 1 + (3*n + 4)*x - (5*n - 21)*x^2 - (11*n - 26)*x^3 - (5*n - 10)*x^4)*v[i - 1] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A200073: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:    1;
  1:    4,     3;
  2:   11,    15,      5;
  3:   29,    56,     35,      7;
  4:   76,   189,    171,     66,      9;
  5:  199,   605,    715,    407,    110,    11;
  6:  521,  1872,   2730,   2054,    832,   169,    13;
  7: 1364,  5655,   9810,   9180,   4965,  1533,   245,   15;
  8: 3571, 16779,  33745,  37774,  25585, 10642,  2618,  340,  17;
  9: 9349, 49096, 112309, 146357, 119168, 62453, 20862, 4218, 456, 19;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (2*n+1+(3*n+7)*x-(5*n-16)*x^2-(11*n-15)*x^3-(5*n-5)*x^4)*P(n-1,x) + (4*x+17*x^2+27*x^3+19*x^4+5*x^5)*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 + 1 + (3*n + 7)*x - (5*n - 16)*x^2 - (11*n - 15)*x^3 - (5*n - 5)*x^4)*P[n - 1, x] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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 + 1 + (3*n + 7)*x - (5*n - 16)*x^2 - (11*n - 15)*x^3 - (5*n - 5)*x^4)*v[i - 1] + (4*x + 17*x^2 + 27*x^3 + 19*x^4 + 5*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-c5713bb0c1f908c6
  • Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-c5713bb0c1f908c6
  • Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-c5713bb0c1f908c6

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023