Real-rooted atlas

A063967 → D+1

nonnegative coefficients

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

Recurrence
\(\left(5 + 5\,x - 3\,x^{2} - x^{3} - 6\,n - 10\,n x + 2\,n x^{2} + n x^{3} + 5\,n^{2} x + n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-10 - 17\,x + 17\,x^{2} + 9\,x^{3} + x^{4} - 4\,n + 28\,n x + 3\,n x^{2} - n x^{3} + 9\,n^{2} - 7\,n^{2} x - 2\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(10\,x + 37\,x^{2} + 37\,x^{3} + 11\,x^{4} + x^{5} + 5\,n + 31\,n x + 36\,n x^{2} + 11\,n x^{3} + n x^{4} + 5\,n^{2} + 6\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
355111
410178136
5204737111116
63811613065141364
7712663892791011711124
813057910489875141452013424
9235121326213069210685519723110320
1042024676202870174463990132225726130832
117444901140602301523659158406931193532529191440
12130895523080457674693265585530568112692714401321269504
132285183246563413840219063417988511860954715173923679485351790080
14397034685136644320550498046539290416942231897922462573648505773812305472
156865649092789697207621247487152562013549618843514243031481183678562476774116700096

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Technical details
Node id
n-f305609b43c3ec0e · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2952378e385b9a17

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 ?

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   2,    1;
   3:   5,    5,    1;
   4:  10,   17,    8,    1;
   5:  20,   47,   37,   11,    1;
   6:  38,  116,  130,   65,   14,    1;
   7:  71,  266,  389,  279,  101,   17,    1;
   8: 130,  579, 1048,  987,  514,  145,   20,   1;
   9: 235, 1213, 2621, 3069, 2106,  855,  197,  23,  1;
  10: 420, 2467, 6202, 8701, 7446, 3990, 1322, 257, 26, 1;
%F Formula
The row polynomials satisfy (n^3-6*n+5+(5*n^2-10*n+5)*x+(n^2+2*n-3)*x^2+(n-1)*x^3)*P(n,x) = -(2*n^3-9*n^2+4*n+10+(7*n^2-28*n+17)*x+(2*n^2-3*n-17)*x^2+(n-9)*x^3-x^4)*P(n-1,x) + (5*n^2+5*n+(6*n^2+31*n+10)*x+(n^2+36*n+37)*x^2+(11*n+37)*x^3+(n+11)*x^4+x^5)*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[Together[(-(2*n^3 - 9*n^2 + 4*n + 10 + (7*n^2 - 28*n + 17)*x + (2*n^2 - 3*n - 17)*x^2 + (n - 9)*x^3 - x^4)*P[n - 1, x] + (5*n^2 + 5*n + (6*n^2 + 31*n + 10)*x + (n^2 + 36*n + 37)*x^2 + (11*n + 37)*x^3 + (n + 11)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 - 6*n + 5 + (5*n^2 - 10*n + 5)*x + (n^2 + 2*n - 3)*x^2 + (n - 1)*x^3)]];
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] = (-(2*n^3 - 9*n^2 + 4*n + 10 + (7*n^2 - 28*n + 17)*x + (2*n^2 - 3*n - 17)*x^2 + (n - 9)*x^3 - x^4)*v[i - 1] + (5*n^2 + 5*n + (6*n^2 + 31*n + 10)*x + (n^2 + 36*n + 37)*x^2 + (11*n + 37)*x^3 + (n + 11)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 - 6*n + 5 + (5*n^2 - 10*n + 5)*x + (n^2 + 2*n - 3)*x^2 + (n - 1)*x^3)); 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.

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-c9969a1cd93f190d
  • 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-c9969a1cd93f190d
  • 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-c9969a1cd93f190d

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. [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.