Real-rooted atlas

A145597 → basis A007318

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(\left(6\,n + n^{2}\right)\,P_{n}(x) = \left(-2\,x^{2} - 3\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(20 + 10\,x + 18\,n + 9\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
22727660
31101657510360
4429858594165152061
516384095382216383152111529
6618818564218401274038225462863728
723256813961156688568035280793888236350136

Show all 15 stored rows

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(6\,n + n^{2}\right)\,P_{n}(x) = \left(20 + 8\,x + 18\,n + 10\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-4355c823139108ab · revision 2 · JSON
Definition source
derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-05e5d91dd5b215be)
Fingerprint
f6-663a3c2c5c184f3a

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:     27,      27,       6;
  3:    110,     165,      75,      10;
  4:    429,     858,     594,     165,      15;
  5:   1638,    4095,    3822,    1638,     315,     21;
  6:   6188,   18564,   21840,   12740,    3822,    546,     28;
  7:  23256,   81396,  115668,   85680,   35280,   7938,    882,    36;
  8:  87210,  348840,  581400,  523260,  275400,  85680,  15120,  1350,   45;
  9: 326876, 1470942, 2813976, 2984520, 1918620, 767448, 188496, 26928, 1980, 55;
%F Formula
The row polynomials satisfy (n^2+6*n)*P(n,x) = -(n^2+3*n+2)*x^2*P(n-2,x) + (4*n^2+18*n+20+(2*n^2+9*n+10)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2)*x^2*P[n - 2, x] + (4*n^2 + 18*n + 20 + (2*n^2 + 9*n + 10)*x)*P[n - 1, x])/(n^2 + 6*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + 3*n + 2)*x^2*v[i - 2] + (4*n^2 + 18*n + 20 + (2*n^2 + 9*n + 10)*x)*v[i - 1])/(n^2 + 6*n)); 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-0c862e94735a3472
  • 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-0c862e94735a3472, e-9e81eec4f5e94153
  • 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-0c862e94735a3472, e-9e81eec4f5e94153
  • 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-0c862e94735a3472, e-9e81eec4f5e94153
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-9e81eec4f5e94153

Proof chains

3 incoming proof steps. 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