A100862 → D

nonnegative coefficients

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

Recurrence

\(\left(-4\,x + 2\,x^{2} + n + 3\,n x - 2\,n x^{2} - n^{2} + n^{2} x\right)\,P_{n}(x) = \left(-n + 7\,n x - 3\,n x^{2} - 2\,n x^{3} - n^{2} + 2\,n^{2} x + 3\,n^{2} x^{2} - n^{3} x\right)\,P_{n-1}(x) + \left(2\,n x^{2} - 4\,n x^{3} + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	3	2						5
3	6	12	3					21
4	10	42	30	4				86
5	15	110	165	60	5			355
6	21	240	645	480	105	6		1497
7	28	462	1995	2660	1155	168	7	6475

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-10d12e855fc37e96 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-b867543e608ceb2b

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:  3,    2;
  3:  6,   12,     3;
  4: 10,   42,    30,     4;
  5: 15,  110,   165,    60,     5;
  6: 21,  240,   645,   480,   105,     6;
  7: 28,  462,  1995,  2660,  1155,   168,    7;
  8: 36,  812,  5208, 11340,  8680,  2436,  252,   8;
  9: 45, 1332, 11970, 39564, 49455, 23940, 4662, 360, 9;

%F Formula

The row polynomials satisfy -(n^2-n-(n^2+3*n-4)*x+(2*n-2)*x^2)*P(n,x) = -(n^2+n+(n^3-2*n^2-7*n)*x-(3*n^2-3*n)*x^2+2*n*x^3)*P(n-1,x) + ((2*n^2+2*n)*x^2-4*n*x^3)*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[(-(n^2 + n + (n^3 - 2*n^2 - 7*n)*x - (3*n^2 - 3*n)*x^2 + 2*n*x^3)*P[n - 1, x] + ((2*n^2 + 2*n)*x^2 - 4*n*x^3)*D[P[n - 1, x], x])/(-n^2 + n + (n^2 + 3*n - 4)*x - (2*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + n + (n^3 - 2*n^2 - 7*n)*x - (3*n^2 - 3*n)*x^2 + 2*n*x^3)*v[i - 1] + ((2*n^2 + 2*n)*x^2 - 4*n*x^3)*deriv(v[i - 1], x))/(-n^2 + n + (n^2 + 3*n - 4)*x - (2*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-0dc5896b91f11a29
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-0dc5896b91f11a29
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-0dc5896b91f11a29
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-0dc5896b91f11a29

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-0dc5896b91f11a29

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
  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-e7942008ad3e1420 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-7078061f6d32a158
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
    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-e7942008ad3e1420 (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.