A274496 → D+1

nonnegative coefficients only rational roots

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

Recurrence

\(\left(-4 - 2\,x + 4\,x^{2} + n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(-6 - 8\,x + 10\,x^{2} - 4\,x^{3} - 3\,n + 15\,n x - 6\,n x^{2} + 4\,n x^{3} + 6\,n^{2} - 4\,n^{2} x + 4\,n^{2} x^{2} - n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-4 - 12\,x - 12\,x^{2} - 12\,x^{3} - 8\,x^{4} - 2\,n - 6\,n x^{2} - 8\,n x^{3} + 2\,n^{2} - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	4	2							6
3	8	4							12
4	12	16	4						32
5	24	32	8						64
6	32	72	48	8					160
7	64	144	96	16					320
8	80	256	288	128	16				768
9	160	512	576	256	32				1536
10	192	800	1280	960	320	32			3584
11	384	1600	2560	1920	640	64			7168
12	448	2304	4800	5120	2880	768	64		16384
13	896	4608	9600	10240	5760	1536	128		32768
14	1024	6272	16128	22400	17920	8064	1792	128	73728

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Technical details

Node id: n-6e657ea7ab79a5f4 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b1032d52426bb000

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): yes

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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1;
  1:   2;
  2:   4,   2;
  3:   8,   4;
  4:  12,  16,   4;
  5:  24,  32,   8;
  6:  32,  72,  48,   8;
  7:  64, 144,  96,  16;
  8:  80, 256, 288, 128, 16;
  9: 160, 512, 576, 256, 32;

%F Formula

The row polynomials satisfy (n^2+n-4+(4*n-2)*x+4*x^2)*P(n,x) = -(n^3-6*n^2+3*n+6-(n^3-4*n^2+15*n-8)*x-(4*n^2-6*n+10)*x^2-(4*n-4)*x^3)*P(n-1,x) + (2*n^2-2*n-4-12*x-(2*n^2+6*n+12)*x^2-(8*n+12)*x^3-8*x^4)*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[(-(n^3 - 6*n^2 + 3*n + 6 - (n^3 - 4*n^2 + 15*n - 8)*x - (4*n^2 - 6*n + 10)*x^2 - (4*n - 4)*x^3)*P[n - 1, x] + (2*n^2 - 2*n - 4 - 12*x - (2*n^2 + 6*n + 12)*x^2 - (8*n + 12)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(n^2 + n - 4 + (4*n - 2)*x + 4*x^2)]];
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] = (-(n^3 - 6*n^2 + 3*n + 6 - (n^3 - 4*n^2 + 15*n - 8)*x - (4*n^2 - 6*n + 10)*x^2 - (4*n - 4)*x^3)*v[i - 1] + (2*n^2 - 2*n - 4 - 12*x - (2*n^2 + 6*n + 12)*x^2 - (8*n + 12)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(n^2 + n - 4 + (4*n - 2)*x + 4*x^2)); 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-240df8f0760fe339
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-240df8f0760fe339
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-240df8f0760fe339

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

preserver_image derivative_polynomial, shift 0 e-240df8f0760fe339

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-ec4171dcd40223e2 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

References

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