A143685 → D

nonnegative coefficients

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

Recurrence

\(\left(-19 - 2\,x + x^{2} + 29\,n - 8\,n x - n x^{2} - 10\,n^{2} + 10\,n^{2} x\right)\,P_{n}(x) = \left(-9 - 504\,x - 65\,x^{2} - 2\,x^{3} + 19\,n + 752\,n x + 49\,n x^{2} - 10\,n^{2} - 190\,n^{2} x\right)\,P_{n-1}(x) + \left(-9\,x - 343\,x^{2} - 47\,x^{3} - x^{4} + 10\,n x + 380\,n x^{2} + 10\,n x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	11	2						13
3	21	42	3					66
4	31	282	93	4				410
5	41	722	1083	164	5			2015
6	51	1362	5973	2724	255	6		10371
7	61	2202	17763	23684	5505	366	7	49588

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-8d7abba03329c950 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-f5ec68e08843c7bd

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:  1;
  2: 11,    2;
  3: 21,   42,     3;
  4: 31,  282,    93,      4;
  5: 41,  722,  1083,    164,      5;
  6: 51, 1362,  5973,   2724,    255,      6;
  7: 61, 2202, 17763,  23684,   5505,    366,     7;
  8: 71, 3242, 39453, 119044,  65755,   9726,   497,   8;
  9: 81, 4482, 74043, 384804, 481005, 148086, 15687, 648, 9;

%F Formula

The row polynomials satisfy -(10*n^2-29*n+19-(10*n^2-8*n-2)*x+(n-1)*x^2)*P(n,x) = -(10*n^2-19*n+9+(190*n^2-752*n+504)*x-(49*n-65)*x^2+2*x^3)*P(n-1,x) + ((10*n-9)*x+(380*n-343)*x^2+(10*n-47)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(10*n^2 - 19*n + 9 + (190*n^2 - 752*n + 504)*x - (49*n - 65)*x^2 + 2*x^3)*P[n - 1, x] + ((10*n - 9)*x + (380*n - 343)*x^2 + (10*n - 47)*x^3 - x^4)*D[P[n - 1, x], x])/(-10*n^2 + 29*n - 19 + (10*n^2 - 8*n - 2)*x - (n - 1)*x^2)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(10*n^2 - 19*n + 9 + (190*n^2 - 752*n + 504)*x - (49*n - 65)*x^2 + 2*x^3)*v[i - 1] + ((10*n - 9)*x + (380*n - 343)*x^2 + (10*n - 47)*x^3 - x^4)*deriv(v[i - 1], x))/(-10*n^2 + 29*n - 19 + (10*n^2 - 8*n - 2)*x - (n - 1)*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-bfab40bb6e39459c
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-bfab40bb6e39459c
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-bfab40bb6e39459c
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-bfab40bb6e39459c

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-bfab40bb6e39459c

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Pascal-(1,9,1) array.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-f64c8958fa0af494 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-a327bf7575a1329d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Pascal-(1,9,1) array.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-f64c8958fa0af494 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.