Real-rooted atlas

A063007 → x ↦ x+1

nonnegative coefficients

Proved for all n: all roots real, in (−2, −1) · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
31318637
4631329020305
5321900930420702641
61683591081905600189025223525
78989379266594060480308708316924213445
8486392396245012285770803950101607763603634321961825

Show all 15 stored rows

Generated b-files: A006442 (row sums), A376467 ?

Computed in your browser ?

Other recurrences

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

\(\left(-1 + n\right)\,P_{n}(x) = \left(-3 - 2\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(4 + 6\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-0444a4f352052985 · revision 2 · JSON
Definition source
derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-e8c1c1b32f6c7532)
Fingerprint
f6-c277ed2a39f7c92c

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,       2;
   3:      13,      18,        6;
   4:      63,     132,       90,       20;
   5:     321,     900,      930,      420,       70;
   6:    1683,    5910,     8190,     5600,     1890,      252;
   7:    8989,   37926,    65940,    60480,    30870,     8316,      924;
   8:   48639,  239624,   501228,   577080,   395010,   160776,    36036,    3432;
   9:  265729, 1497096,  3660300,  5072760,  4358970,  2378376,   804804,  154440,  12870;
  10: 1462563, 9274410, 25951860, 42060480, 43513470, 29801772, 13513500, 3912480, 656370, 48620;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = (n-2)*P(n-2,x) - (6*n-9+(4*n-6)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*P[n - 2, x] - (6*n - 9 + (4*n - 6)*x)*P[n - 1, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = ((n - 2)*v[i - 2] - (6*n - 9 + (4*n - 6)*x)*v[i - 1])/(-n + 1)); 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.

A376467: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:       1;
  1:       3,       2;
  2:      13,      18,        6;
  3:      63,     132,       90,       20;
  4:     321,     900,      930,      420,       70;
  5:    1683,    5910,     8190,     5600,     1890,      252;
  6:    8989,   37926,    65940,    60480,    30870,     8316,      924;
  7:   48639,  239624,   501228,   577080,   395010,   160776,    36036,    3432;
  8:  265729, 1497096,  3660300,  5072760,  4358970,  2378376,   804804,  154440,  12870;
  9: 1462563, 9274410, 25951860, 42060480, 43513470, 29801772, 13513500, 3912480, 656370, 48620;
%F Formula
The row polynomials satisfy -n*P(n,x) = (n-1)*P(n-2,x) - (6*n-3+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1)*P[n - 2, x] - (6*n - 3 + (4*n - 2)*x)*P[n - 1, x])/(-n)]];
Table[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; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1)*v[i - 2] - (6*n - 3 + (4*n - 2)*x)*v[i - 1])/(-n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #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 (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6c2f57d53dbf15b8
  • 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-437a2dd956a9df17, e-6c2f57d53dbf15b8
  • 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-437a2dd956a9df17, e-6c2f57d53dbf15b8
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-437a2dd956a9df17, e-6c2f57d53dbf15b8
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, -1)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-437a2dd956a9df17

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