A081578 → D

nonnegative coefficients

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

Recurrence

\(\left(-7 - 2\,x + x^{2} + 11\,n - 2\,n x - n x^{2} - 4\,n^{2} + 4\,n^{2} x\right)\,P_{n}(x) = \left(-3 - 60\,x - 23\,x^{2} - 2\,x^{3} + 7\,n + 110\,n x + 19\,n x^{2} - 4\,n^{2} - 28\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 43\,x^{2} - 17\,x^{3} - x^{4} + 4\,n x + 56\,n x^{2} + 4\,n x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	5	2						7
3	9	18	3					30
4	13	66	39	4				122
5	17	146	219	68	5			455
6	21	258	735	516	105	6		1641
7	25	402	1779	2372	1005	150	7	5740

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-5d7aa830bd831f2a · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-635a4dd413d7ba68

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:  5,   2;
  3:  9,  18,    3;
  4: 13,  66,   39,     4;
  5: 17, 146,  219,    68,     5;
  6: 21, 258,  735,   516,   105,     6;
  7: 25, 402, 1779,  2372,  1005,   150,    7;
  8: 29, 578, 3543,  7684,  5905,  1734,  203,   8;
  9: 33, 786, 6219, 19524, 24405, 12438, 2751, 264, 9;

%F Formula

The row polynomials satisfy -(4*n^2-11*n+7-(4*n^2-2*n-2)*x+(n-1)*x^2)*P(n,x) = -(4*n^2-7*n+3+(28*n^2-110*n+60)*x-(19*n-23)*x^2+2*x^3)*P(n-1,x) + ((4*n-3)*x+(56*n-43)*x^2+(4*n-17)*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[(-(4*n^2 - 7*n + 3 + (28*n^2 - 110*n + 60)*x - (19*n - 23)*x^2 + 2*x^3)*P[n - 1, x] + ((4*n - 3)*x + (56*n - 43)*x^2 + (4*n - 17)*x^3 - x^4)*D[P[n - 1, x], x])/(-4*n^2 + 11*n - 7 + (4*n^2 - 2*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] = (-(4*n^2 - 7*n + 3 + (28*n^2 - 110*n + 60)*x - (19*n - 23)*x^2 + 2*x^3)*v[i - 1] + ((4*n - 3)*x + (56*n - 43)*x^2 + (4*n - 17)*x^3 - x^4)*deriv(v[i - 1], x))/(-4*n^2 + 11*n - 7 + (4*n^2 - 2*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-fd64a74b37db6ad7
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-fd64a74b37db6ad7
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-fd64a74b37db6ad7
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-fd64a74b37db6ad7

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-fd64a74b37db6ad7

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Pascal-(1,3,1) 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-fba968684828f914 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-86685874cd5f1179
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Pascal-(1,3,1) 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-fba968684828f914 (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.