A172431 → 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

\(\left(2\,n - 18\,n x + 8\,n x^{2} - 8\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(10\,x^{2} - 8\,x^{3} - 2\,n - 26\,n x - 38\,n x^{2} + 24\,n x^{3} + 48\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(x^{2} - 16\,x^{4} - 2\,n x - 16\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	5	10	3						18
3	7	26	22	4					59
4	9	50	81	40	5				185
5	11	82	204	196	65	6			564
6	13	122	415	624	406	98	7		1685
7	15	170	738	1540	1590	756	140	8	4957

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-1f5879be74cbbb08 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2aa7c5ebcb2842d2

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 ?

A377866: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,   2;
  2:  5,  10,    3;
  3:  7,  26,   22,    4;
  4:  9,  50,   81,   40,     5;
  5: 11,  82,  204,  196,    65,     6;
  6: 13, 122,  415,  624,   406,    98,    7;
  7: 15, 170,  738, 1540,  1590,   756,  140,    8;
  8: 17, 226, 1197, 3224,  4675,  3564, 1302,  192,   9;
  9: 19, 290, 1816, 6020, 11375, 12298, 7260, 2112, 255, 10;

%F Formula

The row polynomials satisfy (8*n^3-8*n^2+2*n+(16*n^2-18*n)*x+8*n*x^2)*P(n,x) = (8*n^3-2*n+(24*n^3-26*n)*x+(48*n^2-38*n+10)*x^2+(24*n-8)*x^3)*P(n-1,x) - ((4*n^2+2*n)*x+(16*n^2+16*n-1)*x^2+32*n*x^3+16*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[((8*n^3 - 2*n + (24*n^3 - 26*n)*x + (48*n^2 - 38*n + 10)*x^2 + (24*n - 8)*x^3)*P[n - 1, x] - ((4*n^2 + 2*n)*x + (16*n^2 + 16*n - 1)*x^2 + 32*n*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 8*n^2 + 2*n + (16*n^2 - 18*n)*x + 8*n*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] = ((8*n^3 - 2*n + (24*n^3 - 26*n)*x + (48*n^2 - 38*n + 10)*x^2 + (24*n - 8)*x^3)*v[i - 1] - ((4*n^2 + 2*n)*x + (16*n^2 + 16*n - 1)*x^2 + 32*n*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 8*n^2 + 2*n + (16*n^2 - 18*n)*x + 8*n*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-eadaf2a55771e938
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-eadaf2a55771e938
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-eadaf2a55771e938
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-eadaf2a55771e938

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-eadaf2a55771e938

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Even row Pascal-square read by antidiagonals.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-83282836b5ac538e 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