Real-rooted atlas

A099040 → D+1

nonnegative coefficients

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

Recurrence
\(\left(-4\,n x - 2\,n x^{2} + 2\,n x^{3} - n^{2} + 2\,n^{2} x + 2\,n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-12\,x - 8\,x^{2} + 4\,x^{3} + 4\,x^{4} - 6\,n + 6\,n x + 8\,n x^{2} + 8\,n^{2} + 2\,n^{2} x - 4\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(-8\,x^{2} + 4\,x^{3} + 12\,x^{4} + 4\,x^{5} - 4\,n x + 6\,n x^{2} + 12\,n x^{3} + 4\,n x^{4} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2210416
301632856
408768816188
5007228022432608
60024392896544641920
70002561664262412801285952
800064161660807232294425618192
9000080078402009619072665651254976
1000001605792323206182448640148481024164608
1100000230431744119168180224120832327682048489088
1200000384188801460484049925038082938887168040961443776
1300000062721155845949441293312136192070246415564881924238336
140000008965747258777622149123932160358195216547843358721638412382208

Generated b-files: A073388 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-5f9de13ad69d402e · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ecc5a96e63634e1d

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 ?

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 2,  2;
  2: 2, 10,  4;
  3: 0, 16, 32,   8;
  4: 0,  8, 76,  88,   16;
  5: 0,  0, 72, 280,  224,   32;
  6: 0,  0, 24, 392,  896,  544,    64;
  7: 0,  0,  0, 256, 1664, 2624,  1280,   128;
  8: 0,  0,  0,  64, 1616, 6080,  7232,  2944,  256;
  9: 0,  0,  0,   0,  800, 7840, 20096, 19072, 6656, 512;
%F Formula
The row polynomials satisfy (n^3-n^2+(2*n^2-4*n)*x+(2*n^2-2*n)*x^2+2*n*x^3)*P(n,x) = -(2*n^3-8*n^2+6*n-(2*n^2+6*n-12)*x+(4*n^2-8*n+8)*x^2-4*x^3-4*x^4)*P(n-1,x) + ((4*n^2-4*n)*x+(2*n^2+6*n-8)*x^2+(12*n+4)*x^3+(4*n+12)*x^4+4*x^5)*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 - 8*n^2 + 6*n - (2*n^2 + 6*n - 12)*x + (4*n^2 - 8*n + 8)*x^2 - 4*x^3 - 4*x^4)*P[n - 1, x] + ((4*n^2 - 4*n)*x + (2*n^2 + 6*n - 8)*x^2 + (12*n + 4)*x^3 + (4*n + 12)*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n^3 - n^2 + (2*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2 + 2*n*x^3)]];
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 - 8*n^2 + 6*n - (2*n^2 + 6*n - 12)*x + (4*n^2 - 8*n + 8)*x^2 - 4*x^3 - 4*x^4)*v[i - 1] + ((4*n^2 - 4*n)*x + (2*n^2 + 6*n - 8)*x^2 + (12*n + 4)*x^3 + (4*n + 12)*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n^3 - n^2 + (2*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2 + 2*n*x^3)); 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-86e33cb6427ca24b
  • 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-86e33cb6427ca24b
  • 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-86e33cb6427ca24b

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