A103327 → 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

\(\left(-6 + 9\,x - 2\,x^{2} + 17\,n - 22\,n x + 4\,n x^{2} - 12\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(3 - 18\,x + 37\,x^{2} - 10\,x^{3} + 5\,n + 13\,n x - 58\,n x^{2} + 12\,n x^{3} - 24\,n^{2} x + 24\,n^{2} x^{2} + 4\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x - 26\,x^{2} + 28\,x^{3} - 8\,x^{4} - 8\,n x + 24\,n x^{2} - 16\,n x^{3} + 8\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	1							5
2	15	12	1						28
3	42	77	24	1					144
4	93	336	234	40	1				704
5	176	1089	1452	550	60	1			3328
6	299	2860	6435	4576	1105	84	1		15360
7	470	6461	22308	26455	11830	1995	112	1	69632

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-e1fe4dbf4e7c5546 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a95c31fc8093635f

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 ?

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:   4,     1;
  2:  15,    12,      1;
  3:  42,    77,     24,      1;
  4:  93,   336,    234,     40,      1;
  5: 176,  1089,   1452,    550,     60,      1;
  6: 299,  2860,   6435,   4576,   1105,     84,     1;
  7: 470,  6461,  22308,  26455,  11830,   1995,   112,    1;
  8: 697, 13056,  64532, 116688,  86190,  26656,  3332,  144,   1;
  9: 988, 24225, 162792, 419900, 470288, 238374, 54264, 5244, 180, 1;

%F Formula

The row polynomials satisfy (4*n^3-12*n^2+17*n-6+(8*n^2-22*n+9)*x+(4*n-2)*x^2)*P(n,x) = (4*n^3+5*n+3+(12*n^3-24*n^2+13*n-18)*x+(24*n^2-58*n+37)*x^2+(12*n-10)*x^3)*P(n-1,x) + ((8*n^2-8*n+6)*x-(8*n^2-24*n+26)*x^2-(16*n-28)*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[((4*n^3 + 5*n + 3 + (12*n^3 - 24*n^2 + 13*n - 18)*x + (24*n^2 - 58*n + 37)*x^2 + (12*n - 10)*x^3)*P[n - 1, x] + ((8*n^2 - 8*n + 6)*x - (8*n^2 - 24*n + 26)*x^2 - (16*n - 28)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 12*n^2 + 17*n - 6 + (8*n^2 - 22*n + 9)*x + (4*n - 2)*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] = ((4*n^3 + 5*n + 3 + (12*n^3 - 24*n^2 + 13*n - 18)*x + (24*n^2 - 58*n + 37)*x^2 + (12*n - 10)*x^3)*v[i - 1] + ((8*n^2 - 8*n + 6)*x - (8*n^2 - 24*n + 26)*x^2 - (16*n - 28)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 12*n^2 + 17*n - 6 + (8*n^2 - 22*n + 9)*x + (4*n - 2)*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-26d52319153a064f
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-26d52319153a064f
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-26d52319153a064f
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-26d52319153a064f

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-26d52319153a064f

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-86d927bb13c60f05 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023