Real-rooted atlas

A143495 → reverse

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(1 + x + n x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2134
317917
4112372777
51189717581372
61252056607812431915
713338018904081336772910481
81426444550154212377214197218760814

Show all 15 stored rows

Generated b-files: A005494 (row sums), A143495 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-06404fc73ac5a6a1 · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-322b0642acfc2857

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: 1,  3;
   3: 1,  7,    9;
   4: 1, 12,   37,    27;
   5: 1, 18,   97,   175,     81;
   6: 1, 25,  205,   660,    781,    243;
   7: 1, 33,  380,  1890,   4081,   3367,    729;
   8: 1, 42,  644,  4550,  15421,  23772,  14197,   2187;
   9: 1, 52, 1022,  9702,  47271, 116298, 133057,  58975,   6561;
  10: 1, 63, 1542, 18900, 124887, 447195, 830845, 724260, 242461, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-1,x) - x^2*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[(1 + (n + 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
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] = (1 + (n + 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); 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.

A143495: reversed triangle, offset 3, from the row P1
%e Example
Triangle begins:
   3:     1;
   4:     3,      1;
   5:     9,      7,      1;
   6:    27,     37,     12,      1;
   7:    81,    175,     97,     18,      1;
   8:   243,    781,    660,    205,     25,      1;
   9:   729,   3367,   4081,   1890,    380,     33,     1;
  10:  2187,  14197,  23772,  15421,   4550,    644,    42,    1;
  11:  6561,  58975, 133057, 116298,  47271,   9702,  1022,   52,  1;
  12: 19683, 242461, 724260, 830845, 447195, 124887, 18900, 1542, 63, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 4, with P(3,x) = 1.
%t Mathematica
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 2]], {n, 3, 12}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 2)); v[1] = 1; for(n = 4, N, my(i = n - 2); v[i] = (1 + (n - 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};
T(12)

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-0c475173369de966
  • 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-0c475173369de966
  • 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-0c475173369de966

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 (3)

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

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.