A137320 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	2	8													10
3	4	24	24												52
4	12	88	144	64											308
5	48	400	840	640	160										2088
6	240	2192	5400	5440	2400	384									16056
7	1440	14112	38976	47040	28000	8064	896								138528
8	10080	104544	315168	433216	313600	123648	25088	2048							1327392
9	80640	876672	2834976	4306176	3591840	1741824	489216	73728	4608						13999680
10	725760	8212608	28144800	46315520	43092000	24296832	8467200	1781760	207360	10240					161254080
11	7257600	85029120	306085824	538208000	546708800	346389120	141364608	37171200	6082560	563200	22528				2014882560
12	79833600	964350720	3622031424	6736516864	7359316800	5122381440	2363251968	732002304	150543360	19712000	1486848	49152			27151476480
13	958003200	11891543040	46357429248	90496952832	105153093760	79130937600	40311246976	14185746432	3453525504	571084800	61208576	3833856	106496		392574712320
14	12454041600	158422072320	638321209344	1300080198144	1593232600960	1281069613824	708685065088	276554696704	76813761024	15098603520	2052098048	183386112	9691136	229376	6062977267200

Computed in your browser ?

Technical details

Node id: n-142cb87fd684dcb1 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-6dbaa8a13258baa3

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     0;
  1:     2;
  2:     2,      8;
  3:     4,     24,      24;
  4:    12,     88,     144,      64;
  5:    48,    400,     840,     640,     160;
  6:   240,   2192,    5400,    5440,    2400,     384;
  7:  1440,  14112,   38976,   47040,   28000,    8064,    896;
  8: 10080, 104544,  315168,  433216,  313600,  123648,  25088,  2048;
  9: 80640, 876672, 2834976, 4306176, 3591840, 1741824, 489216, 73728, 4608;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-4*n+4+(4*n-8)*x+4*x^2)*P(n-2,x) + (2*n-3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 4 + (4*n - 8)*x + 4*x^2)*P[n - 2, x] + (2*n - 3 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 4 + (4*n - 8)*x + 4*x^2)*v[i - 2] + (2*n - 3 + 4*x)*v[i - 1]); 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-f2695afea9239794
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-f2695afea9239794
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-f2695afea9239794

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-f2695afea9239794

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-6aa1e3cc7e408461 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

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