Real-rooted atlas

A113413 → 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
\(\left(6 - 15\,x - 11\,x^{2} + 3\,x^{3} + x^{4} - 13\,n + 18\,n x - 8\,n x^{2} + 2\,n x^{3} + n x^{4} + 9\,n^{2} - 5\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3} - 2\,n^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(2 - 3\,x - 93\,x^{2} - x^{3} + 13\,x^{4} + 2\,x^{5} - 7\,n - 24\,n x + 50\,n x^{2} + 6\,n x^{3} - n x^{4} + 7\,n^{2} + 33\,n^{2} x - 5\,n^{2} x^{2} - 3\,n^{2} x^{3} - 2\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(x - 2\,x^{2} - 43\,x^{3} + 17\,x^{4} + 10\,x^{5} + x^{6} - 3\,n x - 14\,n x^{2} + 22\,n x^{3} + 10\,n x^{4} + n x^{5} + 2\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
266113
310209140
4144842121117
5189213472151332
622152330288110181921
7262286788725301562112512
8303201226216019108802102416761
9344282022461655623684135827227118004
10385523114883213790123606482198434230147525
1142692455015528302103498824654106402778420331124536
124684863782555260070866887861045280165423760506361324317
1350102086463988011057019314021787016078477922246204950600391840092
1454120811402596161911823948725391544936963051901272243535463687024212166065

Computed in your browser ?

Technical details
Node id
n-cc69020d4f98197e · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a9254ac097c2de67

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 ?

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:  3,   1;
  2:  6,   6,    1;
  3: 10,  20,    9,    1;
  4: 14,  48,   42,   12,    1;
  5: 18,  92,  134,   72,   15,    1;
  6: 22, 152,  330,  288,  110,   18,    1;
  7: 26, 228,  678,  872,  530,  156,   21,   1;
  8: 30, 320, 1226, 2160, 1910,  880,  210,  24,  1;
  9: 34, 428, 2022, 4616, 5562, 3684, 1358, 272, 27, 1;
%F Formula
The row polynomials satisfy -(2*n^3-9*n^2+13*n-6-(2*n^3-5*n^2+18*n-15)*x-(3*n^2-8*n-11)*x^2-(n^2+2*n+3)*x^3-(n+1)*x^4)*P(n,x) = -(2*n^3-7*n^2+7*n-2+(6*n^3-33*n^2+24*n+3)*x+(5*n^2-50*n+93)*x^2+(3*n^2-6*n+1)*x^3+(n-13)*x^4-2*x^5)*P(n-1,x) + ((2*n^2-3*n+1)*x+(12*n^2-14*n-2)*x^2+(2*n^2+22*n-43)*x^3+(10*n+17)*x^4+(n+10)*x^5+x^6)*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[(-(2*n^3 - 7*n^2 + 7*n - 2 + (6*n^3 - 33*n^2 + 24*n + 3)*x + (5*n^2 - 50*n + 93)*x^2 + (3*n^2 - 6*n + 1)*x^3 + (n - 13)*x^4 - 2*x^5)*P[n - 1, x] + ((2*n^2 - 3*n + 1)*x + (12*n^2 - 14*n - 2)*x^2 + (2*n^2 + 22*n - 43)*x^3 + (10*n + 17)*x^4 + (n + 10)*x^5 + x^6)*D[P[n - 1, x], x])/(-2*n^3 + 9*n^2 - 13*n + 6 + (2*n^3 - 5*n^2 + 18*n - 15)*x + (3*n^2 - 8*n - 11)*x^2 + (n^2 + 2*n + 3)*x^3 + (n + 1)*x^4)]];
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] = (-(2*n^3 - 7*n^2 + 7*n - 2 + (6*n^3 - 33*n^2 + 24*n + 3)*x + (5*n^2 - 50*n + 93)*x^2 + (3*n^2 - 6*n + 1)*x^3 + (n - 13)*x^4 - 2*x^5)*v[i - 1] + ((2*n^2 - 3*n + 1)*x + (12*n^2 - 14*n - 2)*x^2 + (2*n^2 + 22*n - 43)*x^3 + (10*n + 17)*x^4 + (n + 10)*x^5 + x^6)*deriv(v[i - 1], x))/(-2*n^3 + 9*n^2 - 13*n + 6 + (2*n^3 - 5*n^2 + 18*n - 15)*x + (3*n^2 - 8*n - 11)*x^2 + (n^2 + 2*n + 3)*x^3 + (n + 1)*x^4)); 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-79715410c2bf252f
  • 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-79715410c2bf252f
  • 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-79715410c2bf252f
  • 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-79715410c2bf252f

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.