Real-rooted atlas

A232535 → 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
\(\left(n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(3 + 15\,x + 27\,x^{2} + 21\,x^{3} + 6\,x^{4} - 9\,n - 38\,n x - 61\,n x^{2} - 44\,n x^{3} - 12\,n x^{4} + 7\,n^{2} + 28\,n^{2} x + 41\,n^{2} x^{2} + 26\,n^{2} x^{3} + 6\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-3 - 18\,x - 42\,x^{2} - 48\,x^{3} - 27\,x^{4} - 6\,x^{5} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x) + \left(-6\,x - 30\,x^{2} - 60\,x^{3} - 60\,x^{4} - 30\,x^{5} - 6\,x^{6}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
21101223
31216448134
4136200352192781
5155480145617927684552
6178980448092168704307226531
7110517921142434560535044096012288154634
8113630242553610560023654429286418841649152901273
91171480051744278784841984149094415360008519681966085253004
1012107260971526589442562560605696088473607798784380108878643230616751
111253105601716001427712691891220873216403865605013504038600704167772163145728178447502
121300148722882882882880169861126336512015541862425401753627394048018717081673400320125829121040068261
13135120384464464549120038584832173801472523911168107957452815253504001453326336892338176318767104503316486061962064
14140627300722176995737682148352438577152158760960040007761927095058432882088345675245813764194304000137573171220132659235331704123

Generated b-files: A038723 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-df244c0ad53b2f3d · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3908c67b5f7a1c29

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

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   3;
  2: 1,  10,   12;
  3: 1,  21,   64,    48;
  4: 1,  36,  200,   352,    192;
  5: 1,  55,  480,  1456,   1792,    768;
  6: 1,  78,  980,  4480,   9216,   8704,    3072;
  7: 1, 105, 1792, 11424,  34560,  53504,   40960,   12288;
  8: 1, 136, 3024, 25536, 105600, 236544,  292864,  188416,  49152;
  9: 1, 171, 4800, 51744, 278784, 841984, 1490944, 1536000, 851968, 196608;
%F Formula
The row polynomials satisfy (n^2+2*n^2*x+n^2*x^2)*P(n,x) = (7*n^2-9*n+3+(28*n^2-38*n+15)*x+(41*n^2-61*n+27)*x^2+(26*n^2-44*n+21)*x^3+(6*n^2-12*n+6)*x^4)*P(n-1,x) - (3-(2*n-18)*x-(6*n-42)*x^2-(6*n-48)*x^3-(2*n-27)*x^4+6*x^5)*d/dx P(n-1,x) - (6*x+30*x^2+60*x^3+60*x^4+30*x^5+6*x^6)*d^2/dx^2 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[((7*n^2 - 9*n + 3 + (28*n^2 - 38*n + 15)*x + (41*n^2 - 61*n + 27)*x^2 + (26*n^2 - 44*n + 21)*x^3 + (6*n^2 - 12*n + 6)*x^4)*P[n - 1, x] - (3 - (2*n - 18)*x - (6*n - 42)*x^2 - (6*n - 48)*x^3 - (2*n - 27)*x^4 + 6*x^5)*D[P[n - 1, x], x] - (6*x + 30*x^2 + 60*x^3 + 60*x^4 + 30*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^2 + 2*n^2*x + n^2*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] = ((7*n^2 - 9*n + 3 + (28*n^2 - 38*n + 15)*x + (41*n^2 - 61*n + 27)*x^2 + (26*n^2 - 44*n + 21)*x^3 + (6*n^2 - 12*n + 6)*x^4)*v[i - 1] - (3 - (2*n - 18)*x - (6*n - 42)*x^2 - (6*n - 48)*x^3 - (2*n - 27)*x^4 + 6*x^5)*deriv(v[i - 1], x) - (6*x + 30*x^2 + 60*x^3 + 60*x^4 + 30*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^2 + 2*n^2*x + n^2*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, 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-df9504f6b81fedcd
  • Coefficients of one sign. Every nonzero row, n >= 0: 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-df9504f6b81fedcd
  • Hadamard kernel (Malo). Rows P_n, n >= 0, 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-df9504f6b81fedcd

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

No candidate equalities or related sequences.

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