Real-rooted atlas

A145597 → D+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(\left(12\,n - 6\,n x + 6\,n x^{2} + 8\,n^{2} + 11\,n^{2} x + n^{2} x^{2} + 7\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(48 + 12\,x + 12\,x^{2} + 8\,x^{3} + 68\,n + 50\,n x + 24\,n x^{2} + 10\,n x^{3} + 40\,n^{2} + 44\,n^{2} x + 21\,n^{2} x^{2} + 3\,n^{2} x^{3} + 11\,n^{3} + 17\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(12\,x - 8\,x^{2} - 4\,x^{4} + 16\,n x - 6\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} + 9\,n^{2} x - 5\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
22127654
3551357510275
4120483504165151287
5231138623521428315215733
640634028568840034025462824752
76667434260823780024570718288236104652

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-ef8a25fd6c4a173a · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-64ce0542ce087dc8

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 ?

No OEIS matches.

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:    6,     3;
  2:   21,    27,      6;
  3:   55,   135,     75,     10;
  4:  120,   483,    504,    165,     15;
  5:  231,  1386,   2352,   1428,    315,     21;
  6:  406,  3402,   8568,   8400,   3402,    546,     28;
  7:  666,  7434,  26082,  37800,  24570,   7182,    882,    36;
  8: 1035, 14850,  69300, 139590, 133650,  62370,  13860,  1350,   45;
  9: 1540, 27621, 165528, 442860, 588060, 402930, 142296, 24948, 1980, 55;
%F Formula
The row polynomials satisfy (n^4+7*n^3+8*n^2+12*n+(2*n^3+11*n^2-6*n)*x+(n^2+6*n)*x^2)*P(n,x) = (n^4+11*n^3+40*n^2+68*n+48+(3*n^4+17*n^3+44*n^2+50*n+12)*x+(6*n^3+21*n^2+24*n+12)*x^2+(3*n^2+10*n+8)*x^3)*P(n-1,x) + ((2*n^3+9*n^2+16*n+12)*x-(2*n^3+5*n^2+6*n+8)*x^2-(4*n^2+8*n)*x^3-(2*n+4)*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[((n^4 + 11*n^3 + 40*n^2 + 68*n + 48 + (3*n^4 + 17*n^3 + 44*n^2 + 50*n + 12)*x + (6*n^3 + 21*n^2 + 24*n + 12)*x^2 + (3*n^2 + 10*n + 8)*x^3)*P[n - 1, x] + ((2*n^3 + 9*n^2 + 16*n + 12)*x - (2*n^3 + 5*n^2 + 6*n + 8)*x^2 - (4*n^2 + 8*n)*x^3 - (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^4 + 7*n^3 + 8*n^2 + 12*n + (2*n^3 + 11*n^2 - 6*n)*x + (n^2 + 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] = ((n^4 + 11*n^3 + 40*n^2 + 68*n + 48 + (3*n^4 + 17*n^3 + 44*n^2 + 50*n + 12)*x + (6*n^3 + 21*n^2 + 24*n + 12)*x^2 + (3*n^2 + 10*n + 8)*x^3)*v[i - 1] + ((2*n^3 + 9*n^2 + 16*n + 12)*x - (2*n^3 + 5*n^2 + 6*n + 8)*x^2 - (4*n^2 + 8*n)*x^3 - (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^4 + 7*n^3 + 8*n^2 + 12*n + (2*n^3 + 11*n^2 - 6*n)*x + (n^2 + 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-62daa9fe2e3f6ac3
  • 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-62daa9fe2e3f6ac3
  • 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-62daa9fe2e3f6ac3
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-62daa9fe2e3f6ac3

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023